Understanding Water Behavior

38 BASIC PRINCIPI..ES

24. It Ehrenberger: Versuche Iiber die Verteilung der Drucke an Wehrriicken infolge des I1bsturzcnden '.Vassers (Experiments on the distribution 'of pressuresa\ong the f~~e of w(d .. ;; resulting from the impact of the fa.lling water), Die W IMJserwirtschaft, Vienna, vol. 22, no. 5, pp. 65-72, 1929.

25. 'H&rald Lauffer: Druck, Energie und Fliesszustand in Gerinnen mit grossem Gefiille (Pressure, energy, and flow type in channels with high gradients), Was- serkrafl, und Wasserwirtschaft, Munich, vol. 30, no. 7, pp. 78--82, 1935.

26. J. H. Douma: Discussion on Open channel flow at high velocities, by L. Standish Hall, in Entra.inment of atr in flowing water: a symposium,T1'ansactions, American Society of Civil Engineers, vol. 108, pp. 1462-1473, 1943.

LeI!.'

foI.l- Pit' 67/\1 c: ,Il.1 ,f;.-N eN

.2. . h}j c/rct,;.d.i :/'t'--10 c:

^{c ,'" Y . 5 . , . (, <}

W/ibn

.ftu;o 4,4ff luitf&

^Yl

7

^{tUi, rv:5}

^{(id ..}

^t

f? .

d

^tU.

^{,lillJ-' foe-t ~} Y-'.c,tt/).Je1'

. .k.": In ^a CCt·n~l.pdjW

^4t-

/~1{."fp"U.1/f'A(iCL

I _ P ,~,> J: . . .

c. "e

J' .,t:; 'I

HI]~II tt'l"1 ::>..;t;.1".!/f"''' V"'r1.tl.-7 VW' _..:!.~,~" .it.-.:.~(j

~'i._ .:di.l:£nnd~~L4;'1~ -t&.4i-:5 p,ttt'a .

e>J

^{rret ,/·1/0}

^{fall .}

.. ,he. ellA-of /:J~"m ;$ dtre,·r/6..,;tLul

vV (I! ·'n'YII!'. .s~t:rt,

Vr';,t.. cd cbf ^{#.. is}

^/l.

^J'Vi4 v..u ^¢

^{,pver.( e @'we <5}

~y.& o,C;...l."< ~

iJ d ( .§ / /J / ' . I IS

r1.!1 /!.~Ci/ bkrl~)~""'p1)

/,1 (f?)e.

'!, 6

rtf..~' •

"'"'t

CttYD.iJ,.u.:; r;~yo;o (}v'Iv: vI< /.

/u..-",,'

^{fYr..f t!}

Ci.~"Vl/

^r1

f~J t· ^{'1' .}

. ' 7 . . . . ( ' / . " .) • (

ftvtAP. t'k,~e l',., ctylv)

!/YiM..ltU ... (l!':71tdVt)

~-;P.

f '

T>/~~cf Jt#rry.: ^{(IU9A .~) .}

~it..d-(c4blh <

.5q,-'Ui4'V"t

d.c;ot/..,

( . bq: ... e It:;rY:J

CHAPTER 3

ENERGY AND MOMENTUM PRINCIPLES

3-1. Energy in Open-channel Flow. It is known in elementary hydraulics that the total energy in foot-pounds per ponild of water in any streamline passing through 9, channel section may be expressed as the total head in feet of water, which is equal to the sum of the elevation

@ ,

1

_d

Saelioil 0 ~ I

I ^I I

1. __ ~:lum

^{_ _ •}

_1_1. 1.~_

FrG. 3-1. Energy in grad'.lo1ly varied 'open-channel Row.

above a datum, the pressure head, and the velocity head. For example, with respect to the datum plane, the totltl head H at a section 0 conta.in- ing point A on a streamline of flow in a channel of large slope (Fig. 3-1) may be written

V

2

H

=

^ZA

+

dA, cos

8 ⁺ 2;

^(3-1)

where ^ZA is the elevation of pain t A above the datum plane, dA is the depth of point A below the, water surface measured along the channel section, 8 is the slope angle of the channel bottom, and V.{2j2g is the velocity hea~

of the flow in the streamline passing through A.

In general, every streamline passing through a channel section will have

. 39

1

_. r"',

)

I ,

• i

40 BASIC PRINCIPLES ^,

~ different velocity head, owing to· the nonuniform vel~ity distribution

U1 ~ctual fl~w, . Onl~ in un ideal pal'a:lel fl~w of uniform velocity distri- butl.on can .he velocIty head be truly IdentlCal for aU points on the cross sectlOn. In the ~ase of gradually varied flow, however,· it may be assumed, for pr~ctlCal purposes, that the velocity heads for all points on the ch~nnel sectIOn are equa.l , and the el~ergy coefficient may be used ·to correct for the over-all effect of the nonuniform velocity distribution.

Thus, the total energy at the channel section is

V

²

H = z

+

^rl cos () _.

+

^{a --.:.} _{2g (3-2)}

For channels of small slope, 0 = O. Thus, the tots'! energy at the chan ..

nel section is

H = z +d

+

c ; . -P

2g (3-3)

Consid~r !lOW a prismatic channel of large slope (Fig. 3-1). The line r~pl'eSelltlltgJhL~_~~va~ion of the tota.Lhea..cLO-f.Jlow is the lill.&gy_line.

;rhe slope of the hne IS known as the energy g'radient, denoted by Sf.

The slope oLthe water surface is de;noted by Sw and the slope of the channel bottoml by So ⁼ sin () _. In uniform £10'" S - S - S -_. ", J - w - o-Sln. . () According to the principle of conservation· of energy, the total energy head at the upstream section 1 should be eql.lal to the· total energy head at t~e downstream seetion 2 plus the loss of energy h_f between the two

sectIons; or . . .

V;2 . V" .

Zl

+

d^l cos /1

+

^al ~2· = Z2

+

dz COs /1

+

^az ---.l: .L h

. g 2g , 'f (3-4)

This equation applies to parallel or gradualJy varied flow. For a channel

of small slope, it becomes .

. V;

V~

.

ZI

+

^YI

+

. ^al -2 g = Z2

+

^Y2

+

^{a2 -}2g ^+h. f (3-5) Either of these tW? equations is known as the energy equation. Wh~n al

=

.a2 = 1 and hr

=

0, Eq. (3-5) becomes

+ .

VI2 . · . V₂²

Zl _{. .} Yl+ -2 _g = Z2 +y~

+ -

_2g const (3-()) This is the well-known BernoHl,li energy equation.2

. 1 The slo~e is generally defined as tan O. For the present purpose, however it is

defined as sm O. '

.. 2

It ~s .believed th~t this equil-tioni is ascribed to the Swiss mathematician Daniel , \ Berno\llh only

?y

lllf~l'ence, to gjve recognition to his pioneer achievement in

'JI hy.dro1yn~mICs, m ,partlcular the in~roductio'n of the concept of "~ead." Actuall}', tlu~ e~uatlOn was first formulated by; Leonhard Euler a.nd later popularized by Julius

WelSbl\ch [1 J. . . .

, .

)

ENERGY AND MOMENTUM PRINCIPLES 41 -3-2. Specific Energy, Specific energyl in a channel section is defined as the energJi per pound of water at any section of· a channel measured with respect to the channel bottom. Thus, according to Eq. (3:-2) with

Z = 0, the specific energy becomes .

S. = d cos 8

+

^{a 2g} V2 ^(3-7) or, for 8: channel of small slope and a = 1,

E =y+-V!

. ' . 2g

(3-8)

which illdic!l.tes that the specific energy is equal to the sum of the depth of water and the velocity head. For simplicity, the following discussion' will be based on Eq. (3-8) for a channel of small slope. Since V

= Q/

A, Eq. (3-8) may be written E = y

+

^Q2/2gAz. It can.be seen that, for a given channel section and discharge Q, the specific energy in a channel section is a function of the depth of flow only.

When the depth of flow is plotted against the specific energy for a given channel section and discharge, a specific-energy CUTve (Fig. 3-2) is obtained.

This curve has ,.two limbs AGand BG. 'rh~ .lifl.lb AC.~J)proache;3the

,horiz.ontall\-xis asymptoticallx toward the righ.t. The limb

Be

approaches the line OD as. it extends l.lpward and to the right. Line OD is a line that passes through the origin and !ul:,s _ananglfi

Qf

lr~eJlP.ati.QJUiQl!j11 to '!5~!-­

For a channel of large slope, the angle of inclination of the line OD will be different from 45°. (Why?) At any point P on this curve, the ordi- nate represehts the depth, and the: ,abscissa represents the specific energy, which is equal to the snm:6f the pressure head y a.nd the'velocity head V²/2g. .

The curve shows that, for a given specific energy, there are tW9 possible depths,. for instance, the low stage YI and the high stage y~. The low stage is· called the alternate depth· of the high stage, and vice vel'sa. At point G; the specific energy is a n~inimum. It will be proved Inter that this condition of minimllriJ. specific energy corresponds to the critical state of fio.v. Thus, at the critical state the two alternate depths apparently become one, which is known as the critical depth 1/e. When the depth of ;fiowis greater than the critical depth, the velocity of flow is less than the critical velocity for the given! discharge, and, hence, th:e fiow js sub criticaL When the depth of flow is less than the critical depth, the flow is supercritical. Hence, YI is the. depth of a supercritical flow, and 1/2 is the depth .of a subcritical flow. ' '

If the discrarge changes, the specific energy will be changed· aucord- ingly. The

1wo

^curves A' B' and A"

B."

^{(Fig. 3-2)} represent positions of

1 The concept of specific energy was first ir~troduced by Bakhrneteff [21

id

1912.

42 BASIC PRINCIPLES

,the specific-energy curve when the discharge is less 'and greater, respec- tively, than the discharge used for the construction of the curve AB.

3-3. Criterion for a Critical State of Flow. The critical state of flow ' has been defined (Art. 1-3) as the condition for which the Froude number is equal to uni~y. A more common definition is that it is the state of flow at which the specific energy is a minimum for a given discharge.¹ A

r o - - - -y - - - I

j Supercritica I

,Fw. 3-2. Specific-energy curve.

theoretical criterion for critical flow may be developed from this definition as follows:

Since V

= Q/

A, Eq. (3-8), the equation for specific energy in a channel of small slope with a = 1 ^J may be written

E = y + - - , Q2 217A -

Diffe~entiatiilg with respect to y and noting that Q is a constant, dE = 1 _ ~ dA = I _ VI dA

dy gA3 dy gAdy

. . "

(3-9)

The differential water area

dll

near the free surface (Fig. 3-2) is equal to T dy. Now dA/dy = T, and the hydraulic depth D:= A/T; so the

above equation becomes '

dE V2T 'V2

- = 1 - - = 1 - -

dy ' g A , gD

I Th~ concept of critica.l depth bas'ed on the theorem of minimurh energy Wall first

introdl.).ced by BOss [3J. '

)

,

,<

ENERGY AND MOMENTUM PRINCIPLES 43·

At the critical state of ftowthe specific energy is a min~mum, or dE/dy =0. The above equation, therefore, gives

N\':;"

VI _{2g -} D

4 (

2 ,~ 3-10)

This is the criterion for critical flow, which states that at the critical state of flow, the velocity head is equal to half the hydraulic depth. The above eq~ation may also be written

V!...;gJ5

== 1,which means F = 1; this is the definition of critical flow given previously (Art. 1-3).

If the above criterion is to .beused in any problem, the following con- ditions must be satisfied: (1) flow parallel or gradually val:ied, (2) channel 01 small stope, alld (3) energy coefficient assumed to be unity, If the energy coefficient is not assumed to be unity, the criti~al-flow criterion is

(3-11) For a channel of large slope angle 8 and energy coefficient ct, the criterion for critical flow can easily be proved to be

Vl D cos

e

ct 2g

=

2 (3-12)

where D is the hydraulic depth of the wateI: area normal to the channel bottom. In this case, the Froude number may be defined as

(3-13) It should be noted that the coefficient a of a channel section actually

·varies with depth. In the above derivation, however, the coefficient is assumed to be constant; therefore, the resulting equation is il0t absolutely exact.

3-4. Interpretation of Local Phenomena. Change of the state of flow from subcritical to supercritical

or

vice versa occurs frequently in open channels. Such change is manifested in a corresponding ch&nge in the depth of flow from a high stage to a low stage or vice versa. If the change takes place rapidly o'/er a relati velyshort distance, the flow is rapidly varied and is known as a local phenomenon. ' The hydraulic drop and _{~ , , .} hydraulic jump are. the two types M local phenomena, and, may be

described as follows: .

Hydraulic Drop. A rapid change in the depth of flow from a high stage to a low stage will result in a steep depression in the water surface.

Such a phenomenon is generally caused by an abrupt change .in the channel slope or cross section and is known as a hydraulic drop (fig. 1-2).

At the transitory region of the hydraulic drop a reverse curve usually

~~1

(j

{

()

" I,

i.,' 1

I'

j

,I

,

I

\

\

I,

\ '

1

44 BASIC PRINCIPLES

appears, connecting the water surfaces before and after the 'drop.' The point of inflection on the reverse curve marks the approximate position of the cdtical depth nt which the specific energy is a minimum and the flow passes from a "SUbcritical state to a supercritical state.

The free ove'rfall (Fig. 3-3) is a special case of the hydraulic drop. It occurs 'where the bottom of a fiat' channel is discontinued. As the free overfall entel'S the air in the form of a nappo, there will be no reverse curve in the water surface until it strikes some object at a lower eleva.tion. It.

is the law of nature that, if no energy were added from the outside the

,

y v'{ijl,/::~LJ)

'7

f"'~'''1i

/Theore.licOi water surfoce

---"""",-""-,,,,- ==:::t' ~s~~nLP~~e.!...! I~ _ _ _ -'- Actual WQter surface - -- - -'\. _ _ _ _ _ _ _ _ _ ~

E

~

FIG. 3-3. Free overfa.ll interpreted by specific-energy c~rve.

water sl1T~!:\cewould seek its lowest possible position corresponding to the least possIble content of energy dissipation. If the specific energy at an upstream section is E, as shown On the specific-energy curve, it will COl1- ti~u.e to be dissipated on the way downstream and 'Will finally reach a mllllmum energy content E",,,.. The specific-energy curve shows that the section of minimum energy or the critical section should occur at the brink. r~~~~tie less_tha!Lth.~,J~titical depth _because

!Er~he:~ecl'eas.e," ^In deJ2th woul£.!eguire a.TI, increo.se in sgecific energy,

~.pos~ll>l~o.!ll.p~nsating ^external en~,ls sup~lie<!: ^.The theoretIcal water-surface curve of an overfall is shown with a dashed line

in Fig. 3-3. .

It ~hould be reme~bered that the detel;mination of critical depth by Eq. ,c3-1O) or !3.11) IS based on the assumption of parallel flow a.rid is ap.phc~ble only approximately to gradually varied flow. 1'he flQW at the brink E' actually ..llllt:llilinear, fru:...:the curYatl..lli} of .flow is -P..'CQllQl!nclill;

.hen~, the method is invalid 'for determining the critical depth asJ.~

ENERGY AND MOME~TUM PRINCIPLES - 45 .~~h~~ the brin~. T~actual situation i~ that the brink section is thg true section of minimum energy, but it is not the critical section as com- jnite(fEyt'te prineiple based on the parallel-flow assumpti~ Rouse [~l ...

found that for small slopes the computed critical depth is about 1.4 times the brink depth, or Yo

=

1.4yo, and that it is located about 3y< to 4yc

"behind the brink in the channel. The actual water surface of the over- fall is shown by the full line (Fig. 3-3). ,..--

It should be noted that, if the change in the depth of flow from Do high stage to a low stage is gradual, the flow becomes a gradually varied flow

-

^o

-::!~

_ Ii Y o "

"

""

o.tl:

-,

Specific-ene'g)' CUrye Hydraulic jump

"" _c.

<U

'0 Y

Ir.ilial deplh

Specific-·force Culye

FIG. 3-4. Hydro.ulic ju~np interpreted by.specific-energy and specific-force curves.

having a prolonged reversed curve of water surface; t.his.phcnomenoll may

~ Clllled ~ grad'unl hud7'a1t~ic drop and is no longer a local phenomenon~

Hydraulic Jump. When the rapid change in the depth of fiow is from a low stage to

a.

high stage, the result is usually an abl'llPt rise of water

~e. (Fig;. 3-4, in which the vertjc!l,l scale is exaggerated). This local' phenomenon is known a~J.he hydraulic j1tmp. It occurs frequently in a canal below a regulating sluice, at, the foot of a spillway, or

at

^{the place}

where a steep chann!)l slope suddenly turns jlat.'

If thejump is low, that is, if the change in depth is small, the water will not rise obviOtisly and abruptly but will pass from the low to the high stage through a series of undulations gradually diminishing in size.

Such a low jump is called an uooular jtimp. .

When the jump is high, that is, when, the change in depth is great, the jump is called a direct jump. The direct jump involves a relatively large amount of energy loss thro,ugh dissipation in the turbulent body of water in the jump. Consequently, the energy content in the flow after the jump is appreciably less than that before the jump. .

It may be noted that the depth before the jump is always less than the

46 BASIC PRINCIPLES

depth after the jump. The depth before the jump is called the initial . depth y ¹ and that after the jump is called .the .sequent depOt Y2; The initial and sequent depths VI and Y2 are shown on the specific-energy curve (Fig. 3-4). They· should he distinguished from the altern!J.te. A~p~hs

YI and Y2'! whi(J~~e the two nossible depth~ fo!:....the same specific ener~l' _.~~itial and s~lent depths are actual depths before and after a jump in which ~rgy lOss b.E is invoh-ed. In other words, the specific -energy E ¹ at the initial depth Vl is greater tha.n the specific energy

at the sequent depth .y~ by an amount equal to the energy loss AE. II there were 110 energy losses, the initial and sequent depths would become identical with the alternate depths in a prismatic channel.

3-5. Energy in NQnprismatic Channels. In preceding discussions the channel has been assumed prismatic so t.hat one specific-el~ergy curve could be applied to evil sections of the channel. For non prismatic chan- nels, however, the channel section varies along the length of the channel and. hence, the specific-energy curve differs from section to section. This cbm'plication can be seen in a three-dimensional plot of the energy curves along the given reach of a nonpl'ismatic channel.

For demonstrative purposes, a nOllprismatic channel with variable slope is taken as an example, in which

a

gradually varied flow

is

^carried

from a, stibcritical state to a supercritical state: (Fig. 3-5) .. The vertical profile of the channel along its center line is plotted on the Hx plane with the x axis chosen as the datum. For a variable-.slope channel, it is more convenient to plot the total energy head H = z

+

^y

+

V^Z/2g, instead of the specific energy, against the depth of flow on the By plane. For simplicity, the pressure correction due to the slope a.ngle and curvature of flow is ignored in this discussion. An energy line is then plotted on the Hx plane below a line parallel to the x axis and passing through the initial total head at the H axis. The exact position of the energy line depends on the energy losses along the channel. Four channel sections are then selected and four energy curves for these sections are plotted in the Hy planes

~

shown. The initial section 0 is an upstream section in the sUbcritical-flow region. The two depths corresponding to a given total energy H 0 can be obtained from the energy curve. Shice this section is in the subCl'itical-flow region, the high stage yo should be the actual depth of flow, whereas the low stage is the alternate depth. Similarly, the alternate depths in other sections can be obta,ined. In the downstream sections rand 2, the low stages Yl and Y2 are the actual depths of flow singe they are in the supercritical-flow region. The critical depth at each sey- tion can also be obtained from the energy curve at the point of minimum energy. At &ection C the critical flow occurs, and the depth y, is the critical depth. On the H ^x plane, varioUs lines can finally be' plotted;

showing the channel bottom, water surface, critical-depth line, and

1

\

I

I I

I I

i

!

I

I

ENERGY AND MOMENTUM PRINCIPLES 47

alternate-depth line. At the critical section, it is noted that the three lines, namely, the .yater surface, the critical-depth line, and the alter- nate-depth line, intersect at a single point. It is seen' tho.t, in passing through the critical Reotion, t,he water surface entei'S the supercritical- flow region smoothly.

The· three-dimensional plot of energy curves is complicated. The description given here is used only for helping the reader to visualize the problem. In actual applications, the energy C'lrves may be constructed

X

FIG'. 3-5. Energy in a non prismatic channel of variable-slope, carrying gradually va.ried flow from sllbcritical to supen::ritical state.

separately on a number of tWo-dimensional Hy planes for the chosen sections. The data obtained from these curves are then used to plot the water surface, critical-depth line, and alternate-depth line on a two- dimensional Ex plane. For simple channels, the energy curves are not necessary because thi;l critical depth and alternate depths C3.n easily be computed directly,

Eumple3-L ·A rectangular cha.nnell0 ft wide is narrowed down to 8 ft by a con- trMtion 50 ft long, built of straight wa,lls and a horizontal fiooc. If the discharge is . 100 cfB and the depth of Bow is 5 ft On the upstream side of the transition section, determine the flow-surface profile in the contra.ction (0.) allowing no gradual hydraulic drop in the contraction, Ilnd(b) a.llowing a gradual ,hydraulic drop ha.ving its point of inflection a.t the mid-sectioll of the contra.ction. Th!Jrip.tionalloss through the con-

traction is negligible. - .

,

.

!

II

{

\

:l

48 BASIC PRINCIPLES

" , . .

Sol1ttiDn. From the given data; the total'energy in the approaching flow meas~red above the channel bottom i~

u:

= 5

+

{100/(5 ?< 1O)1'/2g = 5.062 ft. This energy is kept constant throughout l,lIC contractIon, since energy losses are negligible. A hori- zontal energy line showing the elevation'of the tobl head is, therefore, drawn on the channel· profile (Fig. 3-6). .

FIG. 3-6. Energy principle applied to a channel contraction (a) without gr:adual hydraulic drop; (b) with gradual hydraulic .(hop. '

The alternate depths for the given tot .. l energy. e .. n be computed by Eq. (3-9) as

follows: . '

" 100' 5.062 = Y

+

2g(by)' or . , - 5 06'> "

+

155.25 = 0

y . ~Y b'.

This is a cubic equtl.tion in which b is the width of the channel. At the entrance sec-' tion, where b = 10 ft, its s61ution gives two positive roots: a low stage 'YI = 0.589 ft, which is the altemate depth; and ahigh stage Y2 := 5.00 ft, which is the depth of flow.

At the exit section ,where 11 = 8 ft, this. equation gives a low ati\ge YI =. 0.750 ftiand a

high stage Y. = 4.964 ft. !

When no gr~dual hydraulic drop is allowed in'the contraction (Fig. 3-6a), the1depth of flow at the exit section.should be kept at the high stage, as shown, The high stages for other interm£ldiate sections are then compute~ by the above equation, whicli giveB the flow-surface ptofile. Similarly, the low stages are computed by the aboye pro- cedure and indicated by the alternate-depth line~ ;

When a grodua:~ hydrauiic drop is desired in the contraction (Fig. 3-6b), theldepth of flow at the exit ~ection should be at the low stage, Since the point of inAection of the drop or 11. critipal section is maintained at th~ mid-section of the c~ntracti<?n, the

,I

I

I !

! ,

ENERGY AND MOMENTUM PRINCIPLES 49

., critical depth at t}Jis section is equal to the total head divided by 1.5 (Prob. 3-3), or 5.062/1.5 = 3.375 ft. By :r::q, (3-10), the critical velocity ts'eqllnl to V, = V3.375g = 10.4.5 fps. Hence, the width of this' critical section should be 100/(10.45 X 3.38) =

2.83 fi. '

With the size of the mid-section determined, the side walls of the contraction can bo drawn in ,with straight lines. The lm~ and high stages at each section are then computed by the equation previously'given, As the flow upstream from the critical section is subcritical, lts water surface should follow the high stage. Downstream from the critical secti6n, the flow is sllpercritical and its Burface profile 'follows the low-stage line.

The criti.::al-depth line is shown to sepai'ate the high from the low stage or the sub- Cl'itical from the stipercritical region of flow. On the basis of Eq. (3-10), the critica.l depth can be computed from the equation

(lOO/by,), y, 2g = '2

or . y = {/lO,OOO

, "gb%

where' b is the width of the channel, which can be measured from the plan,

It should be noted :that the vertical scale of the channel profile is greatly exagger- ated. Furthermore, the outline of the gradual hydraulic drop is only theoretical, based on the theory of parallel flow. In reality, the flow near the drop is more or less curvilinear, and the .actual profile would deviate from th/'! theoretical one.. .

This example also serves to demonstrate ¹¹ method of designing a channel transition (Arts. ll-5 to 11-7), The designer may fit any type of contraction walls he desires to suit a given flow profile, or vice versa, .

3-6. Momentum in Open-channel Flow. As stated earlier (Art. 2-7), the momentum of. thefiow passing a channel section per 'unit time is expressed by pwQV /y, where p is the momentum coefficient, w is the l.mit weight of water in lb/ft^i, Q is the discharge in

cfs,

^{and V} is the mean velocity in fps,

,---?'?>- According to Newton's second law of motion, the change of momentum per unit of time in the body of water in a flowing channel is equal to the

'~l

.

⁰⁽

. resultant of all the external forces that are acting on the body,Applying this principletO

a

channel of large slope (Fig. 3-7), the fplfo~~ing expl:ession . for the momentum change per unit tirnein;the body of water enclosed betv.:een sections 1 and 2 may be written:

J(.. 1

• .;.. "'>l·~)(qt (3-14)

- cit! , . ,

where Q, w, and 1:::' are :as .. previously defined, with subscl'ipts refe1'l'in'g to '

)>1' C(

_, Yn

sectionr-l~nd

2;

^{P! and P2} are the resultants of pressures acting on the two sections; W is the weight of water enclosed between the sections; and F! is the total external force

QL

friE,tion and reslstanc~~ing-,ilong the lLUliace of conta,Qt bet\V'een.the water and the cha;nl)el.The above equa-'

tion .is known as the m;omentum equation,^l

. I · ,

1 The application of the inomentum principle was first suggested by Belanger [5J.

,

r -

j :

50 BASIC PRINCIPLES

For a parallei or gradually varied flow, the values of PI and P2 in the momentum equation may be computed by assuming a hYdrosta.tic distribution of pressure. For a cur:-vilinea.r or rapidly varied flo"l,' how- ever, the pressure distribution is no longer hydrostatic; hence the ~alues of PI and P_z cannot be so computed but must be cOl'rected for the curva- ture effect of the streamlines of the flow. For simplicity, P1 and P2 may be replaced, respectively, by {)!'P1 and f1~'P2' where {:Jt' and {)z' are the correction coefficients at the two sections. The coefficients are referred

Fro. 3-7. Application of the momentum.principle.

to as pressure-dislribution coefficients, Sjnce and P2 are forces, the.

coeffi(Jients may be specificall~r (JaIled force coefficients. It can be shown thu,t the force coefficient is expressed by

{:J' =

1-:: fA

hdA= 1

+ ~ fA

^{cdA (3-15)}

AZ)o AzJ~

where

z

is the depth of the centroid of the wa.ter area A below the free surface, h is the pressure head on the elementary area dA, andc is the pressure-head correction [Eq. (2-9)1, It can easily be seen th'1t pI is

!.Treater than 1.0 for ooncave flow, less than 1.0 for and equai

=~~ to 1.0 f <2.r...1lllJ:allel flow:

It can be shown that the momentum e.quation is similar to the energy equation when ~pplied to certain flow problems. In this case, a gradually varied flow.is considered; accordingly, the pressure distribution in the sections may be assumed hydrostatic, and (1' = 1. Also, the slope of the . channel is ass~ed relatively small.1 · Thus, in the short reach of a

1 If the slope a.!l~le 8 is large, then PI = }iwdl~ cos 0 and P. = ;fW~1 cos O,where ell and el, are the Beptbs of flow section and cos IJ is a. correction factor (Art. 2-10).

, ,a. tI

= j.Ad4

¹

(cd ^A

" . <.l .4

A

>(:i:.;. ! ^{c dA}

/::}.

j'(,h .{-

c)id

JI.

()

I I

.\

ENERGY AND )';IOMENTUJI{ PRINCIPLES

rectangular channel of small slope and wi~lth b (Fig. 3-7), PI ;'~Wb1l1:l.

. and Pz ^{= 7§.wb}yz²

Assume F I = wh/by

51

where h/ is the friction head and ii is the avera.ge depth, or (YI+ Y2)/2.

The discharge ~hrcillgh the reach may be taken as the produot of the average velocity and the avera.ge area, or

Q ~ ~i(Vl

+

^V~)bii

Also, it is evident (Fig. 3-7) that eha weight of the body of water is W = wbfjL

and sin {j

=

Substituting :111 the abo,'e expressions for the corresponding items in

Eq. (3-14) and simplifying, .

L

I

(3-16)1

I •

Co t1r

IC!cr-'T(~ 0 'tl.

1.

t'Yltkfr; ~t·-Z,a:k.~

1.iJ"

~ e""

t

^Jt-l'!'

~.~~?}

i

i

This equation appes,rs to be practically the same as the energy equation (3-5).

The.oretically sp'3aking, however, the two equations not only use dif- ferent velocity-distribution coefficient,s, although these are nearly equal, but al~o invol;e different meanings of the frictiollltllosses.

,II!.

the energy

!ill.!ll!-t!.on, the Item hr mea::;yres t.he intl?rna.l energy dissipat.ed in the whole

~.~s of the water iI!tbe reseh, whereas the item

hi

in the momentum equatirJll measures the losses due to external forces exerted 011 the water by the walls of the channel. Ignoring the small difference between the coefficients a and fJ, it seems that, in gradually varied flow, the internal- ellergy losses are practically identical with the losses due to external forces. In uniform flow, the rate with which surface forces are doing work is equal to the rate of energy dissipation, In that case, therefore,.

a dist,inction between hI and

h/

does not exist except in definition.

The simila.rit,y between the applications of the energy and momentum principles may be confusing: A clear ullderstandingof the basic differ- ences in their constitution is important, de.'lpite the fact that

m

many instances the two principles will produce practically. identical results.

The inherent distinction between the two principles lies in the fact that energ'J is a i5calar quantity whereas momentum is a vectC)r quantitT also·

the

~nergy

equation contains a term for' internal losses, where:s

th~

momentum equation contains a term for external resistance.

Generally speaking, the energy principle offers a simpler and

clea~er

I;

r +~ (colA

1/

z ) '"

I

~. 52 BASIC PRINCIPLES

explanation than does the momentum principle. But the momentum principle has certain advl.I.ntages in application to problems involving high internal-energy changes, such as the problem of the hydraulic jump. If the energy equation is applied to such problems, the unkriown internal- energy loss represented by h, is indeterminate, and· the omission of this term would result .in considerable errors. If instead the momentum equation is applied to these problems, since it deals only with extemal forces, the effects of the internal forces will be entirely out of considerfltion and need not be evaluated. The term for il'ictionallosses due to ex~ernal forces, on the other hand, is unimportant in such problems and can safely be omitted, because the phenomenon takes place in a short reach of the channel mid the efiect due to external forces is negligibl~ compfkreq with the internal losses. . Further discussions on the solution of the hydraulic- jump problem by both principles will be given IELter (Example 3-3).

An example showing the application ·of the momentum principle to the problem of It broad-crested weir is given below.

Example 3-2. Derive the discharge per unlt width ofa broad-crested weir across a rectangular channel.

FIG. 3-8. Momentum principle ~pplied to flow over a broad-cregted weir.

Boilltiun. The assumptions to be made in this solution (Fig. 3"8) are (1) the fric- tional forces Fr' and F," are negligible; (2) the depth y. is the minimum depth on the weir; (3) at the channel sections under consideration there is parallel flow; and (4)

~~~r ,pressure £w'on ~e weir surface is equal to the total hydros~.tic p~.!!.!!re , meD.S\.Ired below the upstreamwatei: surface, or ..

p .. = Hwh[y,

+-

^{(YI - 11,)] =} Hwh(2YI - h)

The accuracy (If the last assumption has been checked ~xperimentally [61. If the momentum equation (3-14) is ·applied to the body of water between the upstream J . .

r ⁰

⁰ 1">'1

e'Y°;t~,- [1""<

a-"1,..""; 7

(~~4li/M

,rgf1lJ,o/,J<.

(){;.:aty/'b .d:./n

Ct"7fd-t.

'c.~ t5 ^{If) f3}

^L

::

, . ' - 'Y)

c'Af:i

j .. /.:: .r-?

/, .ve-.le, i:-~ ^{' r · »( , I} <Y'£.

.1: /)~;..<_. r . . ' _ ..

,:Ad

.rI.r...,~;,;,;",,! f. . ^...u,:u.,

ENERGY AND MOMENTUM PRINCIPLES 53 approach section 1 and the downstream section 2 at themininium depth OIl the top of the weir, the following equation may be written:

qw

(!1. _ !J_)

^{= HWYI' -} MWy2 2 - Hwh(2y, - h)

g Y2 YI . .

where q is the discharge per unit width of th~ weir'. /' Experiments by Doeringsfeld and Barker 181 ·have shown that, on the/average, VI - 11, = 2Y2. III that case the above equation can be simplified ani1 solved for q,

q = 0.433

V2Y (-4-

_h^{) l~ H% I (3-17)}

. ·Y'T / / .

Conslderillg the limit.of h from ,zero to Infinity, this equation variesjfo~'!l'lR.~L tv q = 2.4.6RH. It is interesting to note that the practical range·1lf tfie coefficient to

H~" obtained by actnal observations' is from 3.05 to 2.67. In applying the momentum principle tiJ"this problem, .it can be seen that knowledge of the internal-energy losses due to separation of flow at the entrance and to other causes ·is not needed in the

,analysis. .

3-7. Specific Force. In applying the momentum principle to a short horizontal reach of ^fl. prismatic channel, the extern:al force of friction and the weight effect of water can be ignored. Thus, with (J = 0 and F, = 0 and assuming also {31 = {3z = 1, Eq. (3-1.4) becomes

Qw (V2 -- 171)

=

^{PI - P2}

g

The hydrostatic forces PI and 'P2 may be expressed as PI

=

wi1A.1 a.nd P2

=

wi2A2

where il a~d i2 are the distances of the centroids of the respective wa.ter areas Al and Az below the surface of flow. Also, YI = Q/ Al a.nd V 2 = Q/ Az• Then, the above momentum equation may be written

ir. +

^{ilAl =}

il.... +

^{ZZA2 (Qr //.,-c}

^1-1,)

(3":18)

gAl g.fi2

'The v:llue of the coefficient actually depends on many factors: mainly, the round- ing of the upstream corner, the length and slope of the weir crest, and the height of the weir. Many experiments on bl'oad~crested weirs haye been performed. From several of the wen-known experiments King [7J has interpolated the data and pre- pared tables for the coefficient uncle·r various conditions. A comprehensive analysis including more recent data and a presentation of the results for practical applications were made by Tracy [8J. The well"known experiments all broad-crested weirs are (1) Bl].zin tests performed in Dijon, France, in 188.6 [9]; (2) U.B.D.lV.Bi Cornell tests performed at Cornell University in 1899 by the U.S. Deep Waterways Board under the direction of G. W. Rafter, and U.S.G.B. Corncllle.sts performed·by the U.S. Geo- logical Survey under the direction of Robert E. Horton in 1903 [101; (3) Michigan tests performed at the University of Michigan during H)28-1929 [11]; and (4) Minne- Bola and· Washington tests performed,'respectively, at the university of Minnesota and Washington State University [6]. For some formulas and coefficients of discharge developed in the U.S.S.R., see [12]. For an analytical treatment of the problem, lIee·

'[13].

54 BASIC PRINCIPLES

The two sides of Eq. (3-18) are analogous and, hence, may be expressed for ahy channel section by a general function

.

^"

Qz'

F

- i t

^zA "(3-19)

u4'_

, ' '

This function cOMists of two terms. The first term is the momentum of the flow passing through the channel section per unit time per unit weight of water, and the second is !:.l!.e for.J<.ELrillr .... unit weigb.:Lm w~t.!lr. Since , both terms al~e essentially force per unit weight of water, their sum may , , be ,c~lled the 8PlJcifjE.l(jr.c~.1 ,Accordingly, (3-18) may be expressfld

B

c·

I

~=-:-_. I ( o 45° for () Ez '£lEiE:,

channel of ,...,

zero or small

slope (0] l.bl

3-9. Specific-fofce curve supplemented \~ith specific-energy curve. (a) Specific- energy curve; (b) channel section; (e) specific-force curve. '

as F ¹ = F ^{2. .} This means that the specific forces of section,s 1 and 2 are equal, p.r:ovided t~t the e~rnal forces and the weight effect of water in the..rea<;h between the two s~ctiQns can be igno];ed.

By plottillg the depth against the spec~fic force for a given. channel section and discharge, a ~pecific-jorce CU'fve is obtained (Fig. 3-9). This curve has two limbs AC and BC. The limb AC approaches the horizontal axis asymptotically toward the right, The limb BC rises upward and extends indefinitely to the right. For a given value of the specific force, the curve has two poss(ble depths Yl and Yl' " A13 will be shown later, the two depths constitute the il1it~al and sequent depths of a hyqraulic jump. At point· C on the curve the two depths become one, and the specific force is a minimum. The following argument shows that the depth at the minimum value pf the specific force is eqv,al to the critical

depth.~ ,

1 This has been variously called the "force plus momentum,'~ the "momentum ftux," the "total force;" or, briefly, the "force" pf a stream (see pp. 81 and 82 of [14J).

The {unction represented by Eq. (3:'19) was formula.ted by Bresse [15J for the study of the hYdra.ullc jump to be described in Example 3-3. :

i The conoept of critic!!.1 depth based on tbe theorem of momentum :is believed to ha.ve:been developed by Boussinesq [16).

"

i

i

!

i

ENERGY AND MOMENTUM PRINCIPLES 55 For a minimum yalue of the specific force, the first derivative of F with respect to iJ should bl') zero, or, from Eq. (3-19), '

dF Q2 dA ,d(zA) 0

gP

^{dy T}

--elY

For a change dy in the depth, the correspo,nding change 'd(iA) in the static'moment

of

the water area. about the free surface is equal to [A (2

+

^dy)

+

T(dyP/2] - zA, (Fig. 3-9). Ignoring the differential of higher degree, that is, assuming (dy) ²

=

0, the change in static moment becomes d(iA) = A dy. Then the preceding equation may be written

dF _ Q! dA

+

^{A = 0}

gA2 dy Since dA/dy = T,

Q/

A

reduced to

V,and AfT = D, the above equation may be

(3-10) This is the criterion for the criMeal state of How, derived e~rlier (Art. 3-3).

Therefore, it is proved that the dep~h at the minimum va-lueaf the specific force is the critical depth.I It may also be stated that at the critical state of flow the specific'jorce is a minim.'um for the given discharge.

NQw, comp£1,re the specific-force curve with the specific-energy curve, (Fig. 3-9). For a gi;iell specific energy Ell the specific-energy curve indi- cates two possible depths, namely, a low stage YI in the supel'critical flow region and a high stage yz' in the subcritical flow region.² , For a given

v~lue of F ^1, the specific-force curve also indicates two poosible depths, namely,.ail initial depth lit in the supercritica.l region nncla sequent depth

ljz in the sub critical flow region. It is assumed that the low stage and the initial depth are both equaJ to YI' Thus, the two curves indicate jointly that the sequent depth Y2 is a.lways less than the high stage 1/2'.

Furthermore, the specific-energ_J' curve shows that the energy content E2 for the depth Vi is less than the energy content El for the depth Y2'.

Therefore, in order to maintain a constant value of F 1, the depth of flow may be changed from Yl to Y2 at the price 'of losing a certain amouilt of energy, which is equal to El - E, = I:J.E. One example o( this is the

1 It should be noted that the above proof is based on the assumptidrul of para.liel flow and uniform velocity distribution. However, the concept of !lritico.l depth

is:

^a

genera.l concept, tl:..a.t is valid for aU flows, whether derived from energy or from momentum considerations. This validity has beeJ;!. proved by Jaeger [14,17,18), and the proof is known llS the J a.eger theorem {I9]. '

~ In order to make 2. clear distinction between the sequent depth and the high of tile alternate depths, the sequent depth is designated by y. and the ):ligh sta.ge lit. In some other places in thLs book, however, both are designated by 1/1'

, I

I,

56 BASIC PRINCIPLES

hydraulic jump on a horizontal floor, in which the specific forces before and after the jump are equal and the loss of energy is a consequence of the phenomenon. This will be explained further in the following exam- ple,' It may be noted at this point, however, that the depths YI and

yz'

shown by the specific-energy curve are the alternate depths; whereas the ?epths. Yl and Y2 shown by t.he specific-force curve are, respectively, the mitinl depth and the sequent depth of !1 hydraulic jump. '

! '

Example 3-11. Dedve a relationship b~tween the initial depth and the sequent depth of 8: hydraulic jump on a horizontal floor in a rectangular channel. ' - , Sob/ion. The el:ternal forces of friction and the wet'lH effect of Wil.Wr in the

hydl'~ulic jump o~ a horizontal floor are negligible, because~ the-jump't.;;:kes place in a relatively short distance and the slope angle of the ,horizontal fioor is zero. The' specific forces of sections 1 and 2 (Fig. 3-4), respectively, before and after the jump, can,therefore be considered equal; that is,

(3-18) For ll. recLanguinr channel of width b, Q = V,A, = V.A.., Al = by" A, = bYt, 2,

=

yi/2, and ii' - y~/2. Substituting these relations IlJld F, = V d

v'Uih

in the above equation and simplifying,

(:1!!)' -

_{y , .} ^(2F,'

⁺

¹⁾

(J!J)+

_~ ^{2F,' = 0 (3-20)}

Factoring,

[(1Lt)'

_YI

⁺

_. ^~ _YI

^-

^{2F,'] (~} _YI ^-_'

1) -

⁰

Then, let

(

^{~ V}

+

^{1l! _} 2F • = 0

Y.:/

y, I

TIle solutjon of this quadra.tic equation is

(3-21) . For a. J:!.iv:? ,F'roude ?u~ber FI of the apprQaching flow, the xatio of the sequent. depth to the IlUt1l3l depth IS given by the above equa.tion.

It should be understood that the momentum principle ill used in this solutiou because the hydraulic jump involves a high amount of internal-energy losses whioh cannot be evaluated in the energy equation., '

.The jOilit. UIlB of the specific-energy curve aud the specific-force curve helps to d~ter­

~ne gra.plllcally the energy loss involved in the hydraulic jump for ^II. given appro8.ch- mgflow. For the given approaching depth 11., points P, ana. P,' are located on the spe~itic-force curve and the spec~c~energy curve, respectively (;Fig, 3-4). The point Pi'!gives the initial energy content E ,. Dralv a vertical line, passing through the P?int PI and intercepting the upper limb of the specific-force curve at point P" which gives the sequent depth 1)2. 'Then, draw a horizontal line passing through the point Pa and intercepting the specific-energy curve a.t point P.", which: gives the energy con- tent E, after the j';lmp. The energy loss in the jump is then equal to El - E"

represented by / l E . : '

a-8.Momentum

priIicipl~

Applied to

Nonprisrnati~

Channels. The sp~cifj.c force, like the specific' energy, varies with the snape ofthechannel

' .. ,I

"'i f i \

I

ENERGY AND MOMENTUM PRINCIPLES 57 section.' In applying the mornentur~l principle to nonprismatic channels, therefore,a three-dimensional plot similar to that shown for the applica- tion of the energy principle (Fig. 3-5) can be constructed. For practical purposes, however, this is rarely necessa.ry.

Where there is no interyention 'of external forces or where these forces are either negligible or given, the momentum pdriciple can be applied to its best advantage to problems, such as t.he hydr.alllic jump, that deal

;"!I"':';...~;.tJ,,""'!j12!JiJ."'J="_<".bJ'-JlJ""'''''' th.at..!:~l!!!i~~. jUhe energy hydraulic jump is involve(i.

following example shows how the momentum design of a channel transition in which a

Example 8-4., A rectangular channel Sit wide,.earrying 100 cfs at a depth of 0,5 ft, is connected by a str:oight-wall transition to ^Il. channel 10 it wide, flowing nt.s. depth.

FIG. 3-10. (mnergy and'momentum principles applied to ^II. channel expansion (a.) with hydraulic jump; (b) without hydraulic jump. . . :

o~ 4 n (Fir;. 3~lO). Determine the flow; profile in the transition if th~ frictional loss through tl;\e tra.n&'ition is negligible. If, a hyqraulic jump occurs in the transition,

how C!l.n i~ be eliminated? - : ' ,

S"l'Uti"n~ From the given data, the tptal energy with respect to the channel bot~' tom in the approaching flow is E ~ 0.5

+

[IOO/(O.S X 8))'/2g = 10.207 It, and in the dowiistrea.i:n, _, E - 4,0

+

(100/(4 X 10)Ji/2y _,

.

= 4.097 ft. ,It is appnhnt that this