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File CSPY BUREAU OF RECLAMATION HYI~AULI C LABORATORY

NOT TO BE RF21OVEO FROM FILES

* * * * * * * * * * * * * * * * * * * * * *

,ma~ S~A~S DEPAR~T OF ~HE INTERIOR

BUREAU OF REC~0N

IULDRAULIC LABORATORY REPORT E~O, L x)

F~0N IN STEEP CHUTES ~Tl~i~I SPECIAL ~ E

• 0 SELF-~ION

By R~ Ehrenberger

Translated By Ed~mrd F. ~ilsey, Assistant Engineer

Denver, colorado Ootober 193~

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• t~Aous ~ p e e and I ~ t ~ : . e~e +l:~.sed on "khe %aolt & s t a t i o n t h a t tho

s/~o.tfLe p 'wr l .~ e~' the liqluid As ur iah . ~heoo ~mmmmont8 u ~ u l e ne~ be

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applicable, should they be first calibrated in ~.~ter aerated to various

desrees, for, as the experkments showed, ti:e -.~mter presented different

degrees of aeration at different •points of the cros~ section which would

have to be pre~iously determined. ~efore proceeding with thedeter"

mination of velocities, the complete lay-out of the apparatus and the

general plan of the investi~ation will be described in detailo

The experiments ~re performed on chutes with ~ive different

slopes and with ,four different discharges for each ,slope. .These bottom

slopeslin test series I to V were S - 15.5, 20.6, 32.0, 49.5, and ?6,2

percent. The last value corresponds to the slope of the wastermy at

the Rutz Works. Anabsolute fall ~of 3~5 meters (iI.5 feet) was avail-

able in the laboratory. The len~th along the incline for the smallest

slope (test series I) was 16~9*.ers i(52.5 feet) and for the maximum

slope (test series V) was 5.5 meters (18 feet). In ,,~eneral, those

lengths were entirely ~sufficient to ins;re uniform Dlow at the lower ~

end nf the chute. 0nly in,test series V, at the higher velocities,

did it appear that uniformity was not completely,achieved. .'~enoe, in

this case, extrapolation had to be employed. The disc.harses used for

each of the test series were i0. 20, 31, and ~4.5 second-liters (0-353,

0.706, 1.09, ~Lnd 1.-57 second-feet). Occasior~lly, a discharge as low

as throe second-liters '.(0.-i00 second-feet) was used as a check. The

44.5 second-liter (1.57 second-feet) discl~rGe v~s the maximum attaln-

able under the given laboratory conditions. A bafZle served to quiet

IBy the slope, S, Is meant the tangent of the an~le of incllrmtion

of the bottom of the chute. ~he slne of the angle of inclination had

to be introduced, however, in the final formula for theoretloal con-

s i derati ons.

; b

2

i. ̧ ;

the flow issuing from the high reservoir. The •cross section of the

channel w~s rectangular and in conformity to the dimensions of the Rutz

waste chute (bottom width 2. bmeters or 8.2 feet~ had a width of about

0. 25 meters or 0,82 feet.

A certain problem arose in connection with the exact determination

of the water surface. As a result of :the aeration phenomenon (produced

by air resistance), with the ~naller slopes, the surface w~s much

roughened, and in addition, detached drops of water moved along parallel

to the water surface, while with Greater velocities, a water surface, in

the usual sense of the word, simply did not exist because of the gradt~l

transition from air to water. With the larger discharzes in test series

V, in which surface velocities of 8.5 meters per second (27.9 feet per

second) occurred, the water surface exhibited, as a result of the large

~mount of intermingled air, the same milky-white appearance noticeable

to a greater degree at the Rutz wasteway. Figure 2 shows a schematic

section through a chute. At the top, droplets of w~ter interspersed

through air are first noticed. Below this .layer, there is a layer con-

sistin G of a mixture of air rand water, which in turn covers ~, layer of

water containing individual air bubbles, and finally there is a layer

of unaerated w~ter adjacent to the bottom. With steep slopes (test

series IV and V), the voter-air mixture extended all the way to the

floor o f the chute. From this description, it may be ~athered that the

definition of the term "water surface" is an important one. In what

follows, the depth is def~ned as that height, above ~he bottom, at which

the uppermost layer of.water rdrops rebounds ,with considerable force

against the b~oad side of a flat bar, held in the channel perpendicular

to the direction-of flow. The s~mple apparatus . oonsSrue ted fQr measuring

the water surface permitted moving the bar for determining elevation as

well as length so that both longitudinal and transverse w~ter surface

profiles could be •determined.

3

• S e l f o A e r a t t o n o~ Water

As aiready mentioned, ~the phenomenon of self-aeration is also

severed briefly in the tests. By means of several ~lass windows placed

on the sides of and extending ~to the bottom of the Chute, this phenomenon

could be directly and effect ivel2 observed. A white area indicated the

aerated ~ter, while below this strip, extending to the floor, a clear

portion (air-free) was observed. Bet~on ~h,)se two layers, a more or

less well defined 'Boundary laye#'was noticed markin G the lhuit of the

aeration. If the velocity hea~, determined by means of a ~s~_mple Fitct

tube, held at v~rious heights along a normal to the w~ter surface, is

plotted on coordinate paper, charact~ristic ~or~al velocity head carves

are obtained (figure 7). The elevation o£ t.hat point denoted by tl,

above which the veloclt~j head, ns a result of aeration, decreases rapidly

agrees well, Tot the ~s.ll slopes (test series I-II!), with the elevation

of the previously mentioned , boundary layer." Pitot tube measurements

made along the center line of the chute also gave informs&ion on the

variation of the aeration in the longitudinal direction. Fig~re~ shows

a longitudinal section in which the sloping bottom is shown horizontally.

The heavy curve shows the water surface, and the dashed curve indicates

the position of the boundary layer. The cross-hatched ,part between these

t~o curves represents the water-air mixture. It is seen from this graph

that the aeration takes place gradually and nQt suddenly, as in certain

cases of the hydra.~lic J~mp. Further, ~a welml defined dip in the water

sure'ace is to be observed. This can be explained by the fact that the

depth of ~ater in the upper part of the channel, where no appreciable

aeration is present, corresponds t o an ~increase in ~elocity , whi~e in

the lower part, in spite of the further veloc~hy increase, ~he depth,

as a result of the aeration, co~tinues to increase until a final constant condition is reached.

.Determir~tion of tl~e. surface ~Velocity

by a Photographic Precedes_

In order to ascertain the avera~e velocity in a cross section,

a knov¢ledge of the surface velocity was first necessary. This w~s

afforded by photographin~ a !luminous float, for a short exposure (0.1

to 0.2 seconds), placed in the upper ~end of the chute. These floats

consisted of small match boxes to which pieces of ma~aesium ribbon were

fixed. The luminous magnesium was recorded on ~he film as a white strip

~.hoee real length could be easily found by comparia Z it with sage lines

of k~aovm, length placed everymeter along the sides of the chute at the

same elevation as the existing :',~ter surface. ~tch boxes were chosen

for floats, in order to prevent the ext!nguishin~ of the light source

by the surrounding w~ter spray. Such a float projectin~ relatively

i:i~h above thewator surface encounters a small air resistance, which

will be discussed i later. Evidently, this procedure is primarily adapted

t o a n instant when the float is in.a region o f uniform velocity. Only

with the largest discharges of testseries V, in w~tich absolutely uni-

form stretches did not exist, viers several luminous floats released

simultaneously close behind one another, in order to arrive at the

trend of the surface velocity in the longitudinal direction.

To find the actual t£me of exposure, was mot so simplo as to

measure the path described by the float. The time of exposure had to

be as short as possible (about 0,I second)for the steep chutes, con-

sidertng the hio~h velocities ~and ithe relatively short uniform paths.

At first the time was measured by photographing on the same film a

clocko placed near to the chute and ha~ing a l~minous ~haad movin E over

a black face, and the luminous float. The elapsed time was found by

simple division f~..~m the angular veloo.it~j of .and the angle traversed 'bY

t ho hand (a stoP watch was used in determining the angular velocity),

the movement of the hand .over the ~black face recording well on the film.

5

!/

'v

H~ever, it was shown that the moti'cn .. of 'the ~b~nd, in .spite of .a .b~l-

ance wheel, was not accurately uniform and, further, the elapsed time

• could not be determined with the .required accuracy by simply .using :a

• stop w~toh. For example, if the velocity is to !be determined within

, accuracy '•Of -- 5 ,perce~Gr, the t&~e must be accu v • ratel , 0.005 second, assuming an elapsed tim e of the exposure y measured t;o

o f about~ 0.1 second. Since such an accuracy could no~ be a t ta ined with th is method,

the time had to be •measured in some otherlway" This was afforded ~by

the use ~of an arc 'lamp suppllied with alternating current (25t000 CP;).

The light emitted ,by an arc lamp, as is well known, is not of constant

intensity i but changes regularly from maximum to minimum eor~espondlng

%o the reversals of the poles (equal to twice the frequency). Accord-

ingly, the path of the han~ is recorded not as a ~:~fcrm white image

on a dark background, but a~ a series~of isolated white Lma~es on a

dark background. If now the frequency of the alternating current

amounts to 50 cycles per second, the number of p~le :reversals is i00;

hence, the time between two consecutive positions:of the hand is ac-

curately O. 0 1 second. The total time of exposure is ~obtained from

~.he n~mber of images of the hand. Using this method, it is evidently

unimportant whether the hand moves wlth uniform angular velocity or

not. Since the frequency, in consequence of the different leads im-

posed on the transmission network, at ~ost varies about two percent,

an accuracy in the measurement of the tLme bet~een two successive

positi, ~s of the hands of 0.004 second was achieved, or about /four

percentl. Accurate experiments proved that this percentage in the most

unfavorable tests should be increased perhaps by a small amount as a

result of the difficulty ~in reading the first and ~last positions of

the ~hand. The total probable error can therefore be ~taken as about

five percent ~ for the .7,ost unfavorable conditions. The desired velocity

is given by dividing the length of the path of the float by the time

6

. . r . . . . •

thus ~ determined. The ~surface ~eloolty at a point ~is ~obtained by this

method or, to 'be more accurate, ~the:average over the width of the match

box (about one-seventh~of the %otal~width of the chute). A!knowledge

of the "average surface velocity"~was ~necessar 5, ~for the further anal-

ysis. Since the float evidently did not move accurately in the middle,

the correspondin~ reduction could not ~follow accordlng to a definite,

fixed ratio, but must~be evaluated by considerin~ the ~position~of the

float at a given instant. The ~maverage surface velocities, Obtained

in this vmy are given in the following table (table i) and arc denoted

v • o

It should be mentioned that the term "average Surface velocity,,

thus obtained, is not absdlutely correct because the float has various

depths of iz~.ersion as a result of the different ~degre~.s cf aeration in

a single test. Therefore, the calculated values do not ~ive the exact

surface velocities. ~In thefollo~in~, the assumption is made ithat an

increase in the velocity of the ~float. This~assumption~appears Jus-

tifiable because, as the tests slowed, the water,portion of the water-

air mixture increases rapidly from top to bottom; :therefore water par-

ticles at the elevation of the bottom of the float play ~the chief part

in impelling the float. The elevation of the bottom of the float was

accuratel> ~determined ~by stretching wirus~across the chute at such an

elevt, tion that the match box.coul'd ~float under the wire close ..to but

not touching it..iThe correct elevation of .the bcttom.of the float

was ~Iven with sufficio.ut accuracy by the elevation of the wires above

the bottom and the height of match box. Check calculations for the

elevation of the bottom of the float .on the basis of ~the specific grav-

it-j of-the v~ter-air mixturc~a~d Archimedes principle gave a good

agreement with the direct observations.

P~y

Q

Q~

R

t

t w

t"

tm

tr

V

V

Vo

vm

Nomenclature

a - angle of inclination of the chute.

A = area of -~ . the cross section.

~/~ = specific weight of ~ater

a~TL = apparent specific r~ight of the ~ater-air mixture.

hw = h -velocity head at a point in ~ater.

hWL " v e l o c i t y head a t a p o i n t i n t h e m i x t u r e o f ~ t e r a n d a i r .

II = average of ~, over the entire crosssection.

~:~ = ratio of tl~ volume of water to the volume of the v~ter-~ir

mixture at a point.

PL = ratio o f the volume o f air to the volume o f the water-air

mixture .at a point.

= average pf p,~ over the entire cross section.

= ~,~ " discharge of water only.

" disclmrge offerer and air combined.

= hydraulic radius

[] depth of flow normal t o the bottom of the canal.

= distance of the bottom of the float above the ibottom of the chute.

= depth measured at the edge of the Rutz chutes

= average depth of flew~at a cross section.

- average depth of unaerated flew = PWtm.

= velocity at a point.

=average velocity~over the entire cross section.

- average %urfaee velocity.as measured by the £1oat.

= average velocity computed from the surface velocity.

8

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D e t e r n i n a t i o n of. the A..v.era~

" V e l o c i , a t a Cross Seot!on and t h e Avera e Aera%ion.

As a l r e a d y mentioned in t he foreword, the de t e rmina t ion of the

velocities at any poin t in the cross section meets with fundamental

difficulties as a result of the aeration. Two independent variables

enter at the same time (velocity and aeration). The oaiibration curve

of the instrument is known. The average normal velocity curve (average

velocity over the whole width of the chute at various depths normal to

the bottom) was determined in the following manner8 Since, with the ex-

ception o f test series V, the aeration did not reach all the way to

bottom, it was possible to establish the average normal velocity curve

by direct Pitot tube measurements at least in the unaerated portion of

the flow. The lower part of the curve was obtained by this means. The

mavera~e surface velocityn, Vo , found photo~raphically, gave another

point on the curve at the elevation of the bottom of the float. The de-

sired c,Arve for v and t can be tolerably drawn through the branch rising

from the bottom and through this single point. Only the top portion of

the curve is doubtful. T~is i~ of no great importance in the further

computations, since the amount of water in the ~p layers amounts to o n l y

a small percent. Finally~ a comparison of the discharge computed from

this normal velocity curve and the actual discharge, as will be shown

later, shows a good agreement.

The average velocity at a arose section cannot be set equal t o

the arithmetic mean of the individual values of the velooi~, because

there are different degrees o f a~ratlon at different depths. Before

goin~ into the calculation of this average velocity, a mathematical ex-~

pression for the term "aeration" must be derived. This seems to,be

necessa ry because the product of the c r o s s - s e c t i o n a l a rea and t h e ave r -

age velocity does not ~ive the actual dlsc.har~e when aeration is present. The expression consists o f the inequali~ys

AV> Q w

i0

~i/r•~ ? ••~• • Z • • ~:~ • • • • :• !i ~

.O

~To produce an equal : i ty , a r e d u c t i o n f a c t o r , :1~, the~peroen t of wa te r ,

.by volL~ne, in the total water-~Ir:mixture, must be introduced. Thls gives the equation:

':Or PW ,= - ~ .v . (1)"

?~ere it possible to measure directly,the~discB~rge actually flowing

through anisolated horizontal.strip,-:the v~lues of the aeration in the

different 18yers couldbe .determined directly. ";Let .:A..Qwrepresen t the

discharge .passingt~ough.a horizontal .strip ~Iose area ris ~ A, and let .W!

u denote the ratio:

then

- - m : U

A .A ((2

i t f o l l o w s from equation (i) that

._I A ~ _I

= ' ~ = ~ C3') P w ~ • ' v v

_I

in which v represents the actual velocity taken f~om the average

normal v e l o c i t y curve a t the g iven s t r i P : a r e a " A A.

F o r the purpose-of :ca: l ibrat ' ing the l a y e r s , a shee t -meta l ~conduit

of rectangular cross section vms fabricated to accurately :fit the inside

of the experimental chute (figur e :6). The upstream end of this conduit

was placedat the lower end of and at various heights above ~the :bottom

of the chute so that the flow. as ~it were, was ~cut into two :parts, one

of which flowed through the cond,~t, the other underneath it. The

latter discharge was caught by a wooden :calibration .tank ~of a .oapacIty

.of :about six.eublo meters.. Thus, :it was possible to de, ermine .the values

of A A .and A Qw,of .equation 2 . ~ rThis method should Eive e.esential,ly cor-

rect 'values in-tests w~th :smal~l iohannel Slopes, for experimental .dlffi-

~culties, due to hi~l~ .me!Iocities, .did not enter,. High ,~e~ocities introduce

ii

%

,i~ ~, . .

errors that are not permissible. It w~s firs~ assumed that using :sheet

metal as thin as possible, wouldnot dis~urb the :flew very much. !Iow~

ever, the stiffness of the sheet•metal was not sufficient to prevent

excessive vibrations of the bottom of the conduit. A further difficulty

lay in' accurately determining the quantity of water flowing into the

calibration tank ~in a ~iven~tim~..Were it possible to create ~a ,quiet

horizontal wuter surface in the calibration tank with high velocities

of flew, a further •difficulty would still be ~encountered as a result

of the •large •quantities of water partly aerated dischargin G into the

tank,. Much air would then be in the tank, resulting in water depths

that are too 'large. Consldering,these difficulties the depth in the

tank had to be measured cautiously. This m~thod was only used ~for

checking the tests on the flatter slopes.

In comparison with this method, the use of a Pitot tube ~(without

an ejector) possessed greater possibilities for determining the actual

value of the average aeration. The up.ream leg of the Pitet tube had

• an•opening . of~about two millimeters and could be moved up and down and

crosswise of the chute. To reduce the :incipient, :large pulsations, the

glass riser tubes ;had a relatively i:lar~e diameter (50 millimeters).

Since, as is w~ll known, when a static leg (~irected downstream)~is also

taken into account, the velocity head, h = ~-~ 0 is given:by the differ-

once between the kinetic ~le~ reading and the static leg reading. In

spite of the large damping effect, pulsations persisted. Thereforew:

average readings w~re taken after observin G the fluotuatln~ water

columns for several minutes. Consequently, a complete .Pitet tube trav-

erse .required four or five hours.

Before computing the aeration (expressed by Pw) from the velocity

head, several remarks are in order,. The weight per second of the dis-

charge, A Qw, flowing through a small strip of area, A A can be expressed

in two ways:

JJ'

!i ̧ ::

-e

,Q

71. ;By the ;discharge and the :specific weight, Cthusl G - . A , % % - ~ A v . ~ .

;2, ,'By the ,voltmle .0£ .the w~ter-air -mixture :flowing ,per ;:second and the apparent .specific weight, a~L , thusl

G - A QWL ~'tL = 'AA . v ~ N L "

hence Pw'= - - - ~

• &Cter those ~kor t , p r e l i m i n a r y :remarks the :correct procedure f o r

computin Z the .aeration from the .velocity head should be investigated. The velocity head of the water-air mixture cannot be used for determining

the apparent specific gravity, 'ClVL , directly. "This veloclty head ~is

hi~iL= ~-. 2e> (5)

If the rising .legs of the Pitot tube do not contain a fluid of •specific

gravity, aWL , :but, on the contrary, umaerated water of:sPecific gravity

sly, as is actually,the case, since the air bubbles .escape from the ri.sing

legs in a short time, the foll~ing >relation obtainsi

or ~. ~.~

el' 1~om :equation (2)

•Equating

w

P.

(5 ) and . . (6) , we have

;,Pw ..~ and, finally,

= 2tthl

C61.)

(7~)

• ~ ' i ~ d

! .i

7!

:,1.3

~i~! ~ ~

i '

,i °

Five complete ~Pitot tube traverses were made for each experiment

for flndin~ the average value of the aeration, Pw, over the whole cross

section. Pw denotes this average value. ?icuros 7 and 8 SlL~ the com-

puted values for test series II, with Q-20 liters per second (0.706

second-foot). The.nLeasurements were made at five verticals, and a total

of 40 points v~s ~cvered. The two ~raphs tc ~he left in fisure 7 show

sample normal velocity-head curves, while figure 8 shows sample hori-

zontal velocity-hc~d curves. The avora~,es of all the readin~s~ includ-

ing those not shovnu in the figures for the sake of Clearness, are given

• be the right in fi2ure ~ and thus represent an averase normal velocity-

head curve. With the aid of these cur,yes and the normal velocity curves,

earlier described, the aeration index for a ~iven horizontal layer can

be computed. From equation (7) we have

Fi~,ure ~9 shows curves for both I~W and v. However, it should be noted

that the lower part of the v-curve, in the reo~ion where no aeration

exists as was seen tltrou~h the ~lass windows, ~s fo,And from the follow-

ing equation, which is valid for ordinary ~o~ditionsl

By averaging the pw-eurve, the final averas.e aeration factor, Pw,

for the entire cross section is obtained. The Pw-curves for all tests

have practically the sam~ sl~pe. Aeration ~begins at the surface where

the percentaj~e oC air, PL " (I - Pw) = 100%, and percentage of water,

= 0%.

As the depth increases, the percentace of air decreases rapidly

at first and then cradually ~lecreases to 0 percent at the elevation ,of

the bounda~j layer (see figure2 ). This depth corresponds ~approximately

to the inflexion point of'the corresponding vertical velocity-head ,our~e.

Evidently~ this limit is not sharply defined. Although the llmit,l~v = I~

14

L

i

/

/i, ¸• :~

/

:::

occurred above the :bottom in test series I to I!l, :it v~s Just at the b

bottom in series IV, ~d in ser:ies V the aeration at the bottom v ~ s

Pw = 0.83. ~%veraging the v-curve doeG not give the proper av~raL~e veloc-

ity., • V, for the whole cross ~ection .because of the .various degrees of

aeration at different levels in the flOWo E~ch indi.vi~ua~ veloclty

must be weigl~ted according "to the aeration, Pw~ st the correspondlhg

depth. The mean velec:ity for ~the entire cross :section was computedby

considering narrow horizontal strips three millimeters high. Average

values of v' and p~for each strip were taken from the proper v and Pw curves. The desired V is then computed from

~; ! V ! v ,, ::(.9.)

This computation procedure can be checked by the discharge. :If ~ Q i~s

the discharge through a narrow horizontal strip whose area :is A A, then

AQ=AAv!~.

and the total diso}~rge is

Q .,, z

The discharge computed from this equation a~reos v~l'l with the dis-

charge measured by a weir. in order to .effect a complete agreement in

the disclmrgss, Pw was :computed directly from

AV

and not from the .average of the Pw-values as described heretofore. V

.is the velocity computed according to equation .(9). These Values of

V are shown in table 1 and they do not deviate appreciably from the

previously computed average values of v.

The check me, hod Just described offers the principal means-of

settin G up the correct average vertical velocity curve for test series

V, im which considerable aeration is present ,at the •bottom of the chute.

The bottom velocities v~re therefore determlned bY extrapolating ,aux-

iliazy ,curves f o r v s and :S (for equal discharges),. • ~[~

15

I;

• i k¸"

There is another method for oomputin~ pwwithout including ~he

surface velocity measurements. It is based on the discharge measure-

ments and the average velocity }mad, !~, over theentire cross sec-

tion, obtained by finding the moan of the hw-eurves such as the one

shown to the ri~ of figure 7. Correspondingly, equation 7, w~ich is

valid for any single poLnta can be made to :apply, with close approxi-

mation, to the whole cross section by introducing average values of

the variables over the whole cross section. Thus

V

~om equation (I).

APw

Introducing t:-is equation into (7), .we ~hnve.,

2 2 ( 12 )

The v~luos in column i0 of table I, computed from this .equations in

~en6ral, are Somewlmt smaller than the previous ~p~ obtained by using +

v o. This may be explained, as will ibe shown at the conclusion of this

investigation~ bj. the fact that due to neL, lecting the air resistance of

the float in measurino~ the surface velocity, average velocities values

,were obtained that are too low.

.D

16

~ ~ i,/~ ~

Summary and Compar,.ison of the

Experimental Re su~t s t

In order to eliminate the ~unavoidable errors ~of observation and

to obtain a general idea of the accuracy involved, the following graphs

were prepared. On account 'oi" the limitations of space, o~ly the most

important are reproduced herewith. Figure I0 shows the relation ~between

Vo and Q for the various slopes; figure II shows the ;relations betwmen

t m and tan a for various discharges, Q. In ~both cases a group of curves

are drawn arbitrarily through the plotted ~points. In figure 12 curves

are plotted for :comparing t m and :the computed values of v o. For find-

ing the average depth, t m was plotted a~ainst tan ra rather than ~against

Q, for, primarily on account of the phenomenon of aeration, no coherent

relation was found in the latter ease. Figure 13 is applicable for

finding the elevation, t', of the bo~tem of the float above the :floor

of the chute. Finally, in figure 14 the average velocity, V, for the

entire cross section and the average aeration, Pw, for the whole cruces

section are plotted against the average depth, t m, over the whole cross

section. Table I is a s~-~mary ..of the results.

Analysis of the Results

Thus far, for the sake of simplicity and the fact that e rectan-

gular chute was ~empioyed, the average depth, to, for the entire cross

section has been used rather than the usual hydraulic radius, :R. }h~r-

ever, in order to Generalize the test :results, tmwaS replaced by R, in

the logarithmic ~raphs in figure 15. Five parallel, ~straisht , dashed

lines corresponding to different slopes pass through the plotted :points°

For comparison, the computed values from Rehbook,slrevised and complete

• . _ - .

1Rehbook, Betraahtung ~ber Abfluss, S t a u - und Walzenbildung

bei flelssenden Gew~ssern. (Observations on Discharge, Backwater and

Roller Formation in :Flowing Streams)- Julius Sprlnc~er, Berlin, 1917, p.46.

/ i • . .

form of the Ganguillet-Kuttor formula ;(n = O.010)are given in .the same

figure (dashed lines without points). Prom :the flatter :slope oT these

latter lines as compared with the first set of lines, the -mount' ~f

energy dissipation due to self-aeration is to be seen.

It ~is ,worth mentionin G t~-t the Tntersoct~cns iof the two sets of

dashed lines :fall~0etween approximately 3.00 and 3.!50 meters !.per secomd

(9.8 and ii.5 feet per second). In ~oonsequonce of the smallness of the

.angles of intersection, the position of the intersection pc.Lnts, as

read, :are subject to erroz. The choice of Kutter's n also has a mod-

erate influence on the position of these inters(~ctions. As has already

beenmentloned, if the difficulty of the defining ~"wat.er surface, is

considered, it secms possible that the intersections should lie some-

what lower, namel~, at approximately 2.0 meters per second '(5.1 feet

per second). In spite of this, this graph does show that self-aeration

• begins at a definite velocity,. This veloei~: .may be considered to be

from 2 to 3 meters per second (6.6 to 9,8 feet ~per ~second~) for smooth,

wooden flumes; that is, the ~ulid range of the ~exi~ting velocity formulas

v~ry probably extends only up to this limit. At-some greater velocl, ty

the braking effect of aeration comes into ,play.

i

_Some Supp!ementm~-y Remarks

.on-the Phenomenon of Aeration

The underl~,in~ purpose of the preceding investigation was not ~to

study the causes of aeration, ye t •there are •some interesting facts v~ioh

may contzibute to clarifying its behavior, if ~lass plates areattached

to the end of the wooden:chute so as~o £erman exteneion~ef the two side

~lls, aeration:of the under side of the Jet will be produced, since the

Jet on leavi~ g the bottom of the chute comes in,contact with the:alr. In

order to studythe conditions on the bottom:and at the end Of the chute

18

j - .

•m

Lr

better, pieces were cut out :of the :side walls, extending all the vmy ire

the bottom, and replaced by glass plates. Aeration was produced wi~h

a bottom velocity of 3.00 meters iper second (9,8 feet per second). This

test corresponded to test series I vrlth Q. /~51iters per second (1,57

second-feet). A 'slmi~lar condition should be ~observed In a water jet

discharging from a pipe ~line under pressure. ~.'or this purpose a short,

wooden pipe of square ~seetion (ir.ner dimensions i x I o n.~ was fabri-

~cated,and connected to a pipe line under pressure. The two side walls

of the wooden pipe were ~not carried all the way to the end but, as in

the previous case of the chute, were supplemented ~by 51ass plates which

projected out for some distance iron the end of the pipe. ~°ith this

arrangement it was possible to observe the sides of the flow before and

after it emerged from bhe end of the pipe proper. A~ain characteristic

aeration ~vas observed at the top and bottom ~of the jet with an unaerated

portion bet~veen. If the jet is ,discharged under a lo~er pressure, it

retains its shape, there being no disintegration of the Upper and lo~r

surfaces into individual water drops ~(beL:inning of aeration). As the

velocity ~s gradually increased, the two s~rfaoes first exhibited con-

siderable roughnes s and then decomposed into individual drops of water,

and finally, as the velocity vms [increased still more, a typio~l state

of aeration ~ms produced. By careful observation, that velocZty was

sought at which the first separation of drops of water (beglnnin~ of

aeration) occurred. ~ithouorh the beginning of this condition ~was not

sharply indicated, this velooitylmay be said to be approximately three

meters per second i(9..8 feet per second) which is somewhat smaller ~than

the ~limitin g value determined previously from figure 15. V,%ether the

cause of this difference is %0 be attributed to the fscZ that in a steep

chute the air directly ever the-ater surface is already in motion, while

In the case of the pipe, the jet, ~after discharging from the end of %he

pipe comes iinto contact with quiet air; or is to be attributed ~o the

uncertainty, 5in determining the "water surface • (in steep chutes); ._or

19

~ .

whether other conditions play a part, cannot be answered hera. Finally,

let us turn back for a moment. Aeration, as determined from the ear-

lier investigation, began in every case a% 100 percent !(Pw a O) a~ the

water surface and at a definite depth fell to 0 percent (Pw " I). ?/ith

the aid of p- and v-curves plotted from the results of single measure-

ments, it was attempted to find that velocity up tc which aeration in-

creased or at which air Bubbles were still be±ng absorbed. From figure

9 it is seen that the aeration increases, for example, down to a depth

of 7ram. (0.023 feet). This depth corresponds ~o a velocity of 3.80

meters per second (12.15 feet per second!). If all other such oases are

investigated, values are Obtained which vary between the relatively

narrow limits of ;3.5 meters per second (~1.5 feet per second); and £.5

meters pe r second (14.8 feet per second).; the mosul lies around 4.00

meters per second (13.1 feet per second).

~erifioation and Genera)~,tion | i | •

~,of ~,the Resu~lts ~Found f,or ~he ~iiodel

Chute on .the ~Basis .of ~easurement

o f a n A c t u a l Chute a t the Ruts Wor~-

As a l r eady mentioned a t the beg inn ing , the chute i n s e r i e s V

of the exper iments had the same s l o p e as t h e wasteway a~ the Rutz Works.

Since the completion of this otructure, so,feral measurements l~ve been

made of the surface velocity by!

i. R~melin and Angerer, three measurements in 1913.

2. der Versuchsanetalt f~r Waseerbau (.Vienna !lydraullc

Laboratory) bY order .of t;he Bureau for .the Eleetri-

£1oation of the Austrian State Railways, four

measurements in 1923.

3. der Wasserkraftwerks . A.G (Water Power Plant ComPany,

WAG), five measurements in 192~.

The discharge for the first case was determined from a calibrated

weir; in the last two cases from the inflow and outflow of a reservoir,

20

The storage ~o t ion o f the ~eservo~r ,and in take tumnel s,v~.s not T~ken

i n to cons ide ra t i on . ~.For ~he :determi~at,~on o f ',~he d ischarge ~ .om ~he

,re s e r ~ o ' i r i n t h e sec end ~oa~e, two .oo~ple~e c u r r e n t m e t e r m e a s u r e m e n t s

made in the tailrace of ~:the turbine dra'f~ :tubes ser~e~, and in the third

case, the disoharse ~iasram of ~he ~urbines ,~s used. None too ~,re~t

~an accuracy could be expected in ~ho discharge thus measured. A ~ some-

wh~t greater accuracy ~s a~tainea in the surface velocity measurements.

The depth measurements were made at only one side ,of the Chute in the

second and third oases, ~;hile in the first they ..were made at both s~ides.

P.Smelin ~found that ithe vmter surface in a transverse profile v~s r_ot

horizontal, but w~s dish-shaped, and ,ft~rthcrmore,. the depths measured

at the two banks of the trapezoidal seetlon show~d a difference up ~o

i0 centimeters (0.~28 feet)at a discharze of ~.4 Oubi0 metezs l~er se-

cond (120 second-feet). Likewise. in spite of the s~numetrical arrange °

ment, similar conditions were observed in the z,:odel tests. '.Vith tbiis

circumstance in mind, as well a~ a later one ccncern~ug the average

aeration, it appears that v~itb the exception of the ~data furnished by

R~mel~n, the depth of flow, in general, appears to be too large. It

seems permissible, therefore, to reduce the measured depths, t", to

correspond. Several considerations lead to the following relation for

findin~ the adjusted depth of flow~

t "= 0-8 t "

The difference tn-t amounts to .5-5 cent,.no,era (0,18 feet) for the

maximum case. This value can be a~coepted ~n view of the difficulty in

determinin~ the w~ter surface. The complete resuIts o~" these measure-

meats are given in ta~.~le Ill.

In order to discuss the average cross-sectional velocity in terms

of the float velocity, it is necessary t o find a relation between V .and

v o. E~idently, ihere we are dea~lin~ with ~uesses. R~melin s view is

that there is a considerable difference bet~en these velocities; other~s

think them to be about equal. The model tests of the Rutz wasteway to

21

i ̧ . ~ , )- ~ ' . ~ r . • ~C.,"

i?~;~.~'~ -~. k . " . '

a scale of~ls:lO shov;a probable r a t i o o f - ~ = 0.9. Therefore, th ' is

wasused in a~ll further calculations.

Since the resL~lts of the measurements in table I I~are eTidently

st~ject to ine~itaNle errors o~obsermation, the oorrespo~ding values

of t and V, and V and Q are plotted in figure 18 and :smooth curves

~drawn through ~hepolnts. For practioal :oonsiderations, a curve:~for

R and ,Vwas substituted for a ouzve for t~amd ~. The adJusted~mli~ues

of V and Q for various va~ues of ~ intable III are taken from figure

18. iThe computed,values:of the average aeration, expressed by

AV

are given in the~lasti!ine. If theadJusted values of R:and Vim

table IIIare plotted on loGarithmic ~paper, as in ~igure 15, a straight

line may be dravmtltrough the points obtained. This line fits the

~points ~vell and from it .the effect Of aerat:ion is readilyd~saernible.

Its slope deviates somewhat from the slope of the lines found from the

model tests and this may beattributed ~o neglecting the air resistanoe

of the iLLminous float. The:maximum,deviation oeourswith Q =44.5

liters per second (1.57 second-feet,. If :the deviation of the ~eloclty,

d V, Is~ta~enas a measure of ~the air resistance, then, s~noe ~he air

resistance is proportional to the square of relative velocity betV~en

the;Sloat and the surrounding~air, we have

V -~K V 2

If ~ths ~ine found ~fzom ~he model tests ~(fi~ure~i~ is prolonged UP-

~mrd and the velocity d~fferenees computed :from this line and the line

for the Rutz measurements, a ,relation between A V~andV is obtained.

Y.~hen plotted logarithmioa!,ly, a ~straight line ~ariation is clearly

given. Thetangent of the~angle between this line and the horizontal

axis is ~.6, so :that ~the ~equation of the line becomes~

K v 1-6

Thus it is seon~ that the ve1~city di£ferenoe oaused bY the air

,k! X

i "l

22

- 0

!::-: ~ O

ffl

.~-.0

IE

&

..~

.CM

' 0

'1 |

0

C • e o •

041 ,.llq

.el . I

e o o o i i l l 414

o o , e e

-r c

,.~

.1~

o e , g o , , o

'I

.E

.,,~

C 'C

C C

.~'~ ~ d "

.1~

,:d S . ~ e o o ,o,q

o o o l g o ~ql

.11 : ~

~ 119 oQ I1 |

,--4 B . ,,~1

. I

o e o l . ~ ~ , e l

l a l e l o o e q l

i o .. o e o e ,qNJ ~ ~,eND

..

~ l . o o o e , o o Oo , OO

o iN

: ®

~0

|1 '

' 0

C ~

• . L r

~ o e . o e l l ,o ,

• .~, , t%"

'C( , ~ I , ~ " (

,("w 'u~"

" 4

o o o~ g o -,oe

• C r(Y'

4 ( .v ' ,

e o 04 9 e o 4

I

I q ~ . ~

~ "d

.~t ' m

" ~ .~'~

141 e4l l i e e o

¢

" 4 ' : :

• . e . ~

.IM I i

!

~3

L(I.

o e , @

o 4 e q

~4

o o o i l

e o e 4

e l l , ~ J

.41

o o , o l

e o eel

,-4 . e

o o i ~

o o , OQ

. 4

0

e ~ q ~

4.3

-0

,rl e o t

• e o o~

~d . q

.q

' l f"

,d

o e e q

I

e , O

~ a o

0

14 •

.41

: f , J

IW . W

o

F.0 (D

cll

4

J ,t:~)

. 0

f

e

!

~L

E-o

c~

r~

. . . : -

I

~ Qq

~lff"

Q Qo Q4

4

a

C

' o l ~o

C LI" .eQ ~

L'-

e.o o,

.t

q

~ ela

In

. o

O

~ o o •

G

bo o Q ~

,Q t tb

.O

4 o o ~

e 0 i

. ~ ! ,,

m e o . e l

ltc O

I>1

'resistance !is~approx~ately proportional to the square of the ~average

cress, sectional •velocity. The unLuportant de-~iation in :the exponents

may :be ,explained by the circumstance tl,at the air directly ~over ~hs

water surface is ,net ;at rest but is actually movin~ in the direction

of the water. ~Thus there is "r. complete Justification for maMing~a

small correction .in :the curve for R and V obtained by model experiments

(dashed llnes in :fiKure 15). By doing this, the 'computed va'lue8 of P w

in colunu~ 7of table i show a small decrease, so ti~t .an excellent

agreement is obtained with the values of Pw in column I0, which are

Computed l'rom the average velocity head. The agreement with these

latter values of Pw which are indePendent of the fl~at measurements

and their 'attendant errors, is:a ,further reason for Justifying ~this

correction. The :corrected curves for R and Pw.are .shown on the right-

hand side of fig~-e 15. From these lines, it ,is-seen that the percent-

age of water in'the total water-air ~low decreases slowly.with:an in-

creasing hydraulic radius. The ~.~ore or less complete .scattering of :the

~points relatin~ to *.i:e I~utz waste~my,may be attributed to the :probable

irmccuracies in the disc]mrge measurements. The corrected curves for

V and Q (dashed lines)for the Rutz wasteway, using the ne~ :relations

for R and ~,', and R and Pw, are plotted~by way of comparison in .figure 18. "

, . . ~ . . . '

The ,Two ~.~'~ For~ul=s The two final relations,

V = f, (~, sin a) arc. pw ,, f2 (,:~, sin'a),

which we have been searching fo~, are obtained directly-from the log&~

rithmie graph in figure :1.5. Although the first :fu~ction may be repre-

sented by a single :equation, the second requires two parts, acoording

~to whether sin a is smaller or greater Khan 0.476 (28.5°). :This :limit

seems to correspond !to tl~t .slope at which ithe aeration extends ~all ;the

way to the bottom of the chute. This view is corroborated by directly

27

ebseT,'ving the flew through the glass W i n d o w in test series I,V i(sin a ~'-

0.444)~ in which %his aeration condition was approximately~reproduced,

while ;for test seriesI t o ~III it.~s not the case.

The new formulas (in English units) are,

V = 9 7 R 0~52 sin a 0 ' ' 4

s i n a O' l

The percentage of water ~in the total wnter-airmixture Is ex- ~pressed~by thefollowing two,equatlons! "

-'0.26 Pw "/~ 2R'0'05 sin a forsin a < 0.476

;Pw ~= .30 2"O'05.-sin a "~3"7/~ for ,sin a > 0.476

These equations are valid for artificial.chutes of dressed.wood

and approximately rectangular cross section. They are represented

logarithmically in flgure.19 ;for practical use. They are valid up toe

R- 0.30 m. :(0.98 feet) and sin a - 06707 (a -45o).

It is seen from this graph that within the given valid range an

upper, limiting value for thevelocity, independent of the hydraulic

radius and the :slope, is not yet reached. The ~uestlon, whether the

curve of R and .V with greater R-values a0quires.a :steeper slope and

finally ends vertically, cannot be answered on the:basis :of ithe ex-

perlments made heretofore. It should also ibe mentioned that the com-

puted-velocities for a free overfall ~into'an air-filled space cannot

• b e applied, to our study.wlthout further:ana!y.sls ' since for steep

chutes in the limiting case ~nhen,sin a -1 !(vertical chute), the air

enters at only one side, the.other :throe :sides bein~ protected from

the surrounding air bywalls. As t o ~.the:development,of the aeration,

it is to be emphasized that with e~ual wall ~roughnesa, ~the slope Is the

most important factor, and t h e hydraulic radius and hence the depth are only minor factors.

,26

f

• . ~i. L.., ̧.~•..!.,i~, ' •~ '•' • . . . . •i' ~ ~ '

~.~

h

Finally, attention should be calledtoacircumstancewhich may

be adapted to olarifylng the problem of self-aeration fromanother

point,of vi~v. If the simultaneous ~lues of~tm and V are taken from

theresults era single experiment and plotted on Iogarit.hmic paper,

the graph on the /eft-hand sideof figure 20 is obtained. The

frictional effect o f aeration is clearly s~own in that at the larger

slopes, other conditions remaining constant, the depth suddenly in-

creases with increasing slope. If the average depth~at a cross sec-

tion is m~itiplied by the aeration ratio, Fwo r

t r ~ I~ t

and the,simultaneous values of t r and V are plotted in a slmilar fash-

ion (right-~,~nd side of figure 20), the disturbing aetlon of aeration

is a~ain brought out. It should be remembered that the valuesofV

and P w are taken from the corrected lines of figure 15 and that~the

chute is rectangular in cross seCtlon.

C_onclusions

The foregoing investigation w~sundertaken to olarlfy the prin .

ciples offlovr insteep chutes, and the phenomenon of aeration. The

degree of •self-aeration is denoted by thelpercen t of water, pw, i n

the total water-air mixture. The relation:

Q = A V P w

serves to express the measurements undertaken in a chute, V iis the

average velocity at a cress section, and Pw ~is the averageaeratlon in

the two principal equations I and II o r t h e graphical representation in

figure 19. These equations are valid for artificial •chutes of dressed

wood, with approximately reactangula r cross section. Although no state-

ment of their absolute accuracy can be given, in view of the difficulties

of establishing a single arg~nent for such, they should serve a useful

purpose to the practicing ensineer in designing,new-structures. In the

27 • •r i

interest of the ooiltinuous de,Telopment of the ~k~nowlcdge coaoerning flow

in steep ohucea, let the desire be expressed that the results oi" ve-

locity and discharge measur~nts at all such exi~tln~ structures be

given the widest distribution possible.

I,,̧ ¸̧

lib ~ .,..,.,.~n. .......... ,I~"{~':'.o ~°°~ ~-l-T~- l I t-f~ 7

:1 FIGURE 2"SCHEMATIC CROSS 8ECTON O~THE ~; " ~ - • ,

" ~ • 6 8 0 1 2 -

~ ., THE"OHUTE

t.. 0,04 ~ ! "

I o.o,

o ~ . ~ . o ~o VrLO~'IIY H~'&O,h~t'IN l[l~1 i ~

FIGURE 7 "NORMAL VELOGIT'r -HEAD CURVES

..~- ~_- ..., ~.~.~.--~ .... -,.~ 0.09 . . . . . . . . . . . . . . . . . . . . . . . .,,..

°°°1 t . . . . FIGURE B" HORIZONTAL-VELOCITY

- o.o,; HEAl] (~URV[S

o ~ - - "~ - "~ " : -~ ' - " : " "v'." 0,0,, . ..................... • j... :

o" CC,~,T~,a- .-:-~_.~.#.-'-- , j ~ '~ I ' , . . , > " l i I

~E,,o- , . . . . o - - E s ~, i ! ;

_,o~//;~t_.~ I~ '~ ........ • . . ~ - - ~ l ! k - ' - . . . . . ~ . IL . . cs , . "

- - ,., . . . . . ,e t . . . . ~ lU~

2 _ _ _] ~ / I ~ I ; '~

t~n(:[;~N pERCE~'r ~0 ,,m- ~O I~ 8[COl~9-Fi[i[r 2.0

FIGURE I I - CURVr. S FOR FIN{) NG THE AVPRAGE FI@URE IO'~'CURVES,.FOR FINO|NG CROSS-SECT$OH,(~L DEPTH THE Av|rRAGE~.S.URFACE.VELOgiTy

~, -. ~ - N¥O-Z9

~.~ . " - .

~ - . - ~ - - - - : _ ~ _ _ , _ . _ ~ L - '

:J

r r

I~l~ QeI~;' Illt~., r . . '

~ ~ ' ~ . . . . Z . O.;~06.- ' ' v . . . . . .

V o IN FE[T PER

~.~l-I'k I-I- 2O 25

SL~ONO

' F I G U R E 1 2 - R E L A T I O N " BETWEEN fm A N D V o

,~ .~.~.~ .~.~,~,~.~

d • 1.67 ~;EG.'FT. *'~ I I - 0.1! ) = I - t ~

F::,-;:I:: . . . . . . . . . . . . . . . . /

.-,-,, .o,,:~:~i : J ~ " ~ : ,, .......... .?:I-.M' 7/:IE~-i:

°"'"""~'1~.~ . ........... ......... " '"- "- ]~-i- .... ~ I ~ - - 0,4 0.50~ ~ 7 0 8 ~ 0 4 I I~ IG 210 ?4

P~ V IN F [ (T PF.R ~;[CON~

F I G U R E I 4 - . R ( L A T I O N S B E T W E E N I T H tr A V E R A G E CROSS-SECTIONAL V E L O C I T Y ' A N D DEPTH AND THE

",AVe.RAG F" C R O S S - S E C T I O N A L : A E R A T I O N ANO DEPTH

0.9

O.', *-* . . . . . . * - . -~ / 0,4 )-* . . . . . . ~---

~,, L I - I ,-!/

V IN F(L'T PER 6(COND Pm

F I G U R E 1 9 - L O G A R I T H M I C , R E P R ( S E N T A T I O N OF THE TWO F O R M U L A S . V . f . ( f l ; S ) A N D P w - f ) ( R , S )

Q* LS7 $rc.* FI[

o* h09 8£C.1~.

Q*~.?06 S(~.. r lr.

~0,365 S(¢, Fr.

O.ZO ~;':~.c:-rrJ ' ° "~1 I I I I I

O.lO

:,. o.,,,I.-,...,,.-,-.,. Qq.S7 SE~-FT.

• 0.04 Q,<~70~I~..~.. irT"

00~

V II~il lrlLrT V I l I flEET .

F I G U R E ~ 0 - ~,OMPARISON OF T H F RELATIONS

BETWEEN ?m AND V. AND f r A N D V I N , W H I C H

t r *, Pw f m

0 1 0 - ~ ' . - L, • _ - : ~ : ~ ~ ~ : _ - " ; . ~ .,

.

0 0.1 F 0.3 0.4 0,~ 0*6 0.7 O.e ton C~

• FIGURE 13*•DETERMINATION OF" THE' ELEVATIOr~ . f : .OF THE FLOAT, BOTTOM

. c ° ~ ? I . - L L ~ , . . . . ~L.-J . . . . . ' . , .

°I-ltl;~-k~o-L////PI-II "c ~ l i i , , 3 @ L . . L , ' ~ • • . _.~ i;noL*0,444 Jo . ,

o, I L i :: o.,l: l. l,. K//kd.2 :12 .... .+. ' l i

'0.,o .I.]: . ' ..I: i : . ~. ~i.~,II "~o~- , / - - " : : " - H ' I I

.:o°:Z_ - -~ : . , - : ? . . . . . . .

o.~i ,#,t-i-.--I-.i?hl .-. :WFt--i-H,d-t I

0,'0~ *0 1~ 20 3 0 - 4 ~ !~O~ :~ lO 03 114 GS ~0 V IN Fr [~ " PIER IS(GONO Pw

~I~:UR ~" 1 5 " I N F L U E N C E OF T t ' I E - H Y D R A U L I G R A D I U S R.ON V . A N O . C ~ I ' p w

O IN ~.J~ND* FLEET 0,8 ZS 50 7~ ~O q~ tSO,..r ~-~

" ca : , .... ; ~'J "I • ~ )

I J ,, * f o

. I t .-'~"~.<LV*!l~; ,- ]_ 0,4 ~ . . ~

• q[ G3 "1" ";-- 6" * ]

o : ~,_

o 4- o ~ , 0 f :

• V iN FI[I[T pER IEC01iD

.~ l ISULf i l A¢CORDCNO TO ~" nUtmil*n

VWi,~JIlO~lt ( V*4n no Hydl'mdi¢ I,,.o b )

~ 8 0

I .170

o

1,o ilo

F I G U R E . I G * RF.,SULTG OF M I ~ U I U R E M E N T 8 O N . T H E RUTZ , W A R T [ W A Y

~I'41,-

H Y D - 1 9

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