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B-Jump In Sloping ChannelIwao Ohtsu a & Youichi Yasuda aa Dept. of Civil Engineering , Nihon University, College ofScience and Technology , Tokyo, JapanPublished online: 19 Jan 2010.

To cite this article: Iwao Ohtsu & Youichi Yasuda (1990) B-Jump In Sloping Channel, Journal ofHydraulic Research, 28:1, 105-119, DOI: 10.1080/00221689009499150

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Discussion

B-JUMP IN SLOPING CHANNEL RESSAUT HYDRAULIQUE DU TYPE B DANS UN CANAL A FORTE PENTE Wllli H. Hager, Vol. 26, No. 5, 1988, pp. 539-558

A. Discussors: IWAO OHTSU, Associate Professor, Dr. Eng., Dept. of Civil Engineering, Nihon University, College of Science and Technology, Tokyo, Japan YOUICHI YASUDA, Research Associate, Dept. of Civil Engineering, Nihon University, College of Science and Technology, Tokyo, Japan

The writers have systematically investigated basic characteristics of the jump in sloping channels under a wide range of slopes (0° g 6 < 55°), and clarified the condition for the formation of the jump [10]. In this paper, the flow conditions observed by the author are discussed on the basis of the velocity decay for 9 = 45°, also the relationship between the upstream and downstream depths of the jump for 8 > 22° ~ 23° is presented. Further, this relationship is compared with the author's data (0 = 45°),

1 Description of flow condition

The flow conditions of the transition from supercritical to subcritical flow in sloping channels have been classified on the basis of the velocity decay (Table 3) [10]. In this table, h2, is the end depth of the jump in case the toe of the jump is located at the channel-junction, h2 is the sequent depth of JV,(/Z2 = N\12^I%F\ cos 6 + 1 - l)) and hd is the downstream depth.

0 = 4 5 *

1 . 0

0 . 8

Ü . ° - 6

v t o . 4

0 . 2

0 . 0

Sr.bol

• 0 e

F | 9 . 3 9 . 4 9 . 1

h 4 / h 2

1 . 0 6 1 . 4 4 1 . 7 7

t*'*l A. 9

1 3 . 8 1 7 . 4

fo / N l 6 3 . 0 7 6 . 0 9 5 . 0

t%/tc 0 . 0 7 8 0 . 1 8 2 0 . 1 8 3

v<! / v l 0 . 0 8 9 0 . 0 6 5 0 . 0 5 5

rN \

X\\ N o-=«-o_._ 0 . . ._ 0 B_

1 . . . . 1

o o 1

5 0 1 0 0 1 5 0

_

Fig. 15. Variation of t/m/v, with x'lN1(hi.lh2<!liilhi£ 1.6~1.

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Table 3. Classification of flow conditions

slope e Classification of flow conditions

o* < eg

2 1* - 2 2 ' h 2 » / h 2 ( ^ i ) S h d / h 2 < h t / h 2

TypeB-l(B-jump)

h t / h 2 < h d / h 2 D-jump

-W> h 2 , / h 2 £ h d / h 2 S I . 6 — 1 . TypeB-1 (B-jump)

/ W »

2 2 ' - 2 3 "

< 0 ^ 5 5 '

1.6 —1.8 < h d / h 2 g 3 . 0 —3.2 TypeB-2(B-jump)

hd / h 2 > 3.0 —3.2 the formation of jump is not recognized.

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For 0 > 22° ~ 23°, the flow conditions change with hA\hj as follows: a. h1,jh2<Eh^h1^ 1.6-1.8

As shown in Figs. 13d-g and 14d-f, the remarkable surface eddy is formed partly in the sloping and partly in the horizontal channel. Fig. 15 shows the velocity decay for 6 = 45°. In this figure, Um is the maximum velocity (Fig. 16a), x' is the distance from the beginning of the transition zone along the channel bed (Fig. 16a), /s is the length along the channel bed from the beginning of the transition zone to the channel-junction and /(, is the length of the jump along the channel bed (Fig. 16b). As shown in Fig. 15, Um is affected considerably by the channel-junction, and an acceleration zone for £/mappearsjust below the channel-junction. In this case, the remarkable surface eddy is formed, also Um decays sufficiently in a short distance. Accordingly, the formation of B-jump is recognized (this B-jump is called Type B-l (Table 3c)). The end section of B-jump (x' = /,,) is determined as the first section where Um changes substantially with x', and for x' > /„, Um changes very little in the flow direction.

(b) B-j ump Fig. 16. Definition sketch.

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4 5' Sy.lxjl

• u ©

F | 9 . 1 9 . 1 9 . 2

h d / h 2

1 . 8 4 2 . SO 3 . 2 1

e, / N , 1 9 . 9 3 1 . 3 4 4 . 3

fo / N , 1 0 5 . 0 1 3 0 . 0 1 7 0 . 0

ts /!o 0 . 1 9 0 0 . 2 4 1 0 . 2 6 1

»d / v 1 0 . 0 S 3 0 . 0 3 9 0 . 0 3 0

5 O x N ,

1 O O 1 5 O

Fig. 17. Variation of 1',,,/v, with x'/A'1(1.6-1.8<ft(1//ï2<3.0-3.2).

b. 1 .6- l . 8 < A d / / f 2 ^ 3 . 0 ~ 3 . 2 As shown in Figs. 13b, cand 14b, c, a large eddy is formed partly in the sloping and partly in the horizontal channel. However, a remarkable surface eddy is not observed. Fig. 17 shows the velocity decay for 9 = 45°. As shown in Fig. 17, Um is affected by the channel-junction, also the distance for the decay of Um is longer than that for h2Jh2 ^ hA\h2 ^ 1.6 ~ 1.8. In this case, the formation of B-jump is recognized (this B-jump is called Type B-2 (Table 3d)) because Um

decays sufficiently. c. / ; d / / / 2 >3 .0~3 .2

As shown in Figs. 13a and 14a, almost no surface eddy is observed. Fig. 18 shows the velocity decay for 0 = 45°. In this case, Um does not decay sufficiently, and the change of Um is small in the flow direction (Fig. 18). Accordingly, it is difficult to determine the end section of the jump, also the formation of the jump can not be recognized (Table 3e).

1 . 0

0 . 8

U, 0 . 6

v j 0 . 4

0 . 2

0 . 0

e = 4 5* Srabol

• F l

9 . 1

h < / h 2

4 . 4 2 f , / N ,

6 0 . 4 v 4 / v , 0 . 0 2 2

_L J 5 0 1 0 0 1 5 0

N ,

Fig. 18. Variation of C/m/v, with x'/^/,(/?d//;2>3.0~3.2).

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Thus, in the case of 8 > 22° ~ 23°, the formation of B-jump is recognized for h2Jh2 ^ hd/ h2<, 3.0 -3 .2 (Table 3). In the case of 6 <^21°~ 22°, if the downstream depth is small, a remarkable surface eddy is formed partly in the sloping and partly in the horizontal channel, and Um decay sufficiently in a short distance. From this, the formation of B-jump (Type B-l) is recognized (Table 3a). Whereas, if the downstream depth is large enough, a remarkable surface eddy is formed on the slope, and Um decays sufficiently in a short distance. Accordingly, the formation of D-jump is recognized (Table 3b).

2 Length of B-jump

If the length L of the jump is interpreted as the length of the zone within which the energy dissipa­tion is accomplished, the following general expression has been derived [11], [12]:

,/'(L///L, //,///, (=/?), Shape) = 0 (17)

where ML is the head loss in the jump and "Shape" is the shape of the channel. In the case of B-jump, it is considered that lQ is proper for the length of the jump. Accordingly, equation (17) can be expressed as:

(18)

Symbol

O 6 = 8 .

© 9 = 14

0 6 = 1 8 .

9 e = 2 2 .

® 0 = 2 4 .

© 9 = 2 6 .

© 6 = 2 8 .

0 e = 3 o . ® 0 = 3 5 .

• e = 4 5 .

Q 0 = 5 5 .

0 *

. 0 '

5 *

0 '

0 *

0 *

0 '

0 '

0 °

0 °

0 °

11 I I I I I I 1 — I — I — I 0 0 . 5 1 . 0

Hi

Fig. 19. Length of B-jump ( : equation (19)) (/s//„^0.4-0.5 for 0<21°~22°).

f(lolHL,HLlHu0,hdlh2 or /s//0) = 0

2 o r

l o V

\

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Analyzing the experimental data in accordance with equation (18), the following equation is obtained (Fig. 19).

log,o (/0///L) = - 1.71//L///, + 1.58 (0.2 < //L///, < 0.: (19)

In this case, /0///L and HL/Hl are obtained indirectly by substituting the experimentally obtained F\, 0, ljNu /id/yV; and /0/7V] into the following equations.

In h 'olK Hi H\ — HA F\ , , , , \i\ 1 hfl I

-^cos0(\-l(h<iN])2) + - - - f + — t a n ö 2 cos 6 TV] TV,

HL= _ ( ^ / ( W ) ) 2 + 2{hójN{) sec g //, ^ F,2 + 2 + 2(//A |) sec 6» tan (9 + 2 tan2 0

(20)

(21)

Equation (19) is identical to the equation for the free jump case [11], [12]. As the length of the roller is assumed to be proportional to the length of the jump, the following relation for the length of the roller can be derived:

./(/r///L, HjH^rj), 0, hijh1 or /s//0) = 0 (22)

U

3 0

2 0

1 O

O

_1 I I I 1 1 1 1-

0 . 5

Hi

« 1

1 . O

Fig. 20. Length of roller for 0 = 45° ( : equation (19)).

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where /r is the length of the roller along the channel bed. Analyzing the author's data in accordance with equation (22), a clear correlation can be seen (Fig. 20). As shown in Fig. 20, the length of the roller determined by the author tends to be the same as the length of the jump (solid line: equation (19)) determined on the basis of the velocity decay.

3 Relationship of hydraulic quantities governing B-jump

3.1 Application of momentum equation Selecting the zone of the jump (Fig. 16b) as a control volume, equation (23) can be derived from the momentum equation, which is applied in the sloping direction. In this case, the wall shear stress is neglected, and the momentum coefficients at the sections I and II are taken as unity.

Pi - Pü cos 6 + W% sin 6 + (WH - FH) sin B = - q{vA cos 6 - v,) (23)

where v, is the mean velocity at the toe of the jump, vd is the mean velocity at the end of thejump. Regarding the weight of the water W{= Ws + WH) in thejump, equation (24) is obtained con­sidering the surface profile.

W = [WS] + \WH\ = 2

IN? tan 6 + k' N cos 8

+ h , ) l k'iht + h^-l)] (24)

where k' is the coefficient which corrects the difference between the actual weight and the weight of a straight-line surface profile (Fig. 16b: broken line), / is the horizontal length on the sloping channel in B-jump, Lj is the horizontal length of the jump and h} is the depth at the channel-junction under the assumption of the straight-line surface profile. h, can be written as follows:

Ni h. = [l tan 9 + —

cose» + \ hA - / tan 6 +

Ni

cos 6 (25)

By expressing in dimensionless form, equation (25) can be written a follows:

N 1 /

+ — tan 0 ] + cos 6 Ni

I tan 6

\N N

( -

1 cos

\ 9)

cos

k-6 }N -tan 6

(26)

where

-4+ i> 1 / tan 6.

N Ni \ COS0/N

Pi and Pd (the pressures at the toe and the end of the jump) can be written as follows:

Pi=- N? cos hi (27)

Considering the curvature of streamline just below the channel-junction, FH (surface force on the horizontal channel in the jump) can be written as follows:

FH = rWH (28)

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where r is the ratio of the actual pressure on the horizontal channel in the jump to the hydrostatic pressure. Substituting equations (24), (27) and (28) into equation (23), and expressing in dimensionless form, equation (29) is obtained.

k' I

I

JVJ / \COS

1

- + - 1 tan 6 + 7F\ + 1 + tan2 0 + k'(\ - r) M - + —

M N] cos tan 0 tan A',

+ IF, cos 9 = 0

where h-jN\ is expressed by equation (26). Regarding k', the following relation is obtained by dimensional considerations.

k' = ƒ ( / ! , fl.Aa/M)

(29)

(30)

Analyzing the experimental data in accordance with equation (30), the following equation for k' can be derived (Fig. 21).

fc, = 1 + 10-<2.8ta„*+0.74) ( 3 1 )

3.2 Pressure on horizontal channel in B-jump Fig. 22 shows the bed-pressure on the horizontal channel (broken lines) and the surface profiles (solid lines) in B-jump. As shown in Fig. 22, as 9 becomes large and the downstream depth hdjh2

approaches 1, the bed-pressure becomes considerably larger than the hydrostatic pressure just below the channel-junction. This is explained by the result that the curvature of streamline is large near the channel-junction.

1 . 2 , -

1 . 1

1 . O 0 . 0 0 . 5 1 . 0

t a n 6

Fig. 21. Ratio k' as a function of tan 6 and F] (-

Sysbol

3 O •

e © (D Q ©

Fj = 5 Fj = 6 Fj = 7 F t = 8 F1 = 9

F i = 1 0

Fj = 1 1 F i = 1 2

J I 1 . 5

equation (31)).

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N 3 0

2 0

1 0

Pb 1 w N j

e 4 5 *

F l 9 . 3

h d /h-Z 1 . 0 6

L j / N X

6 4 . 0 r

1 . 0 4 9

5 0 1 0 0 N l

N P b

3 0

2 0

1 0

1 w N j e

4 5 * F l

9 . 4 h d / h 2

1 . 4 4 L j / N i

7 3 . 0 r

1 . 0 1 9

5 0

X

1 0 0 N l

h P b

t*l w N , 5 0 r

4 0

3 0

2 0

1 0

0

e 4 5*

Fl 9 . 1

h d / h 2

1 . 7 7 L j / N t

9 1 . 2 r

1 . - 0 0 5

5 0 1 0 0 N l

h d P b w N e

4 5 * Fl

9 . 1

h d / h 2

2 . 5 0

L j / N ,

1 2 2 . 0 r

1 . 0 0 2

5 0 1 0 0

X

Ni

Fig. 22. Bottom pressure profiles on horizontal bed and surface profiles (B-jump). ( : surface profile; : pressure distribution).

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By measuring the pressure on the horizontal bed in the jump, r can be obtained. Regarding r, the following relation is obtained by dimensional considerations.

r=f{e,FhhA\h2) (32)

Arranging the experimental data in accordance with equation (32), the following equation for r can be derived (Fig. 23).

1 + 10 0.6-1 + - —+01 [.mill/,, (33)

3.3 Relationship between upstream and downstream depths of B-jump By substituting equations (26), (31) and (33) into equation (29), the following relationship can be obtained.

A(/olNlJlNuhdlNuFu0)=O (34)

Whereas, from equations (19), (20) and (21), the following relationship can be obtained.

fiihlNi, IINU Ad/M, 9,Fi)=0 (35)

By eleminating /„//V, from equations (34) and (35), the relationship between the upstream and downstream depths of B-jump (equation 36) can be derived (Fig. 24: solid lines).

f{llNuh&\NuFh8) = 0 (36)

As shown in Fig. 24, the experimental data (including Hager's data for /jd//)2^3.2) agree with this relationship (equation (36): solid lines). In this figure, the author's data are plotted using IjM] = Zj//V, — l/y2. In addition, the broken line is obtained under the following assumptions: (a) the surface profile in the jump is a straight line (k' = l), (b) the pressure on the horizontal channel in the jump is hydrostatic (/• = l). In the case of 6 >30°, the difference between the solid line and the broken line is recognized as shown in Fig. 24. From this, the relationship considering k' and r is adequate. Furthermore,

Synbol

©

© • -■

Q:

- i —

6 = 1 4

0 = 2 6 .

9 = 4 5 .

6 = 5 5 .

^ 1®

0

0

0

0

J

2 . 0 3 . 0

Fig. 23. Ratio r as a function of hijh1 and 6 (- equation (33)).

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e = 5 5 . o '

F ] = 1 0

F , - S N X

Pi =4 \ X S \

^^t^W^ ■', i

4*-^

9 : O; • e

O

(b)

O • e

Ti -4.5—5.4 F i - 6.5-7.4 F» " 8.5-9.4

(d) Fig. 24. Relationship between upstream and downstream depths of B-jump.

a. the experimental data by Ohtsu and Yasuda b. the experimental data by Ohtsu and Yasuda c. the experimental data by Hager d. the experimental data by Hager

- 1 0 = 1 1

in the case of' 6 > 22° ~ 23°, it is confirmed that h2,\N\{l\N\-*ti) is larger than h2/N](h2lNl = (V8/7!2 cos 6 + 1 — l)/2). This is explained by the results that the bed-pressure is considerably larger than the hydrostatic pressure just below the channel-junction, and that the curvature of streamline is larger near the channel-junction.

4 Relative energy loss

The relative energy loss //L///i(= rj) =f(0, F\, hdlh2) can be obtained from equations (21), (36) (Fig. 25: solid lines). As shown in Fig. 25, the experimental data (including the author's data for 6 = 45°) agree with this relation (solid lines), also the value of rj has a maximum at hAjh2 « 1.1 ~ 1.2. From this, it is confirmed that the value of rj for hAjh2 » 1.1 ~ 1.2 is somewhat larger than A-jump case [hA\h2 = h2Jh2).

5 Conclusion

The relationship between the upstream and downstream depths of B-jump for 9 > 22° ~ 23° has been obtained on the basis of the momentum equation considering the surface profile and the bed-pressure just below the channel-junction, also it has been confirmed that this relationship agrees with the author's data. Furthermore, it has been shown that the length of the roller (for B-jump) can be expressed by equation (22).

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O = 2 6 . O* 0 = 3 5 . O' o r F l

' F , ' F ! ' F , ' F t

; F . ' F !

= 1 = 9 = 8 = 7 = 6 = 5 = 4

0

o r

1 . o

' © 0 0

N.

\

1 1 L.

+• h d = h z t

. i . . .

$

S 5~S

-e^^

<ï~~ ~~-Q-Q>

1 .

1 . O 2 . O

F ) / F ,

/ / F l '//*! y/*i / / F , / / F I

= 1 0 = 9 = 8 = 7 = 6 = 5 = 4

( b )

1 . O 0 = 5

Hi

H 1

O . 5

h©

) . O

1

(d)

F l ' F , ' F X

' F , ' F , ' F , ' F !

= 1 = 9 = 8 = 7 = 6 = 5 = 4

0

O : F | - i t

2 . O

h 2

Fig. 25. Relative energy loss for B-jump. a. the experimental data by Ohtsu and Yasuda b. the experimental data by Ohtsu and Yasuda c. the experimental data by Hager d. the experimental data by Ohtsu and Yasuda

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References

10. OHTSU, I., YASUDA, Y. and AWAZII, S., Hydraulic jump in sloping channels. Annual conference at Japan Society of Civil Engineers, 44th, 1989 (in Japanese).

11. OHTSU, I., Free hydraulic jump and submerged jump in trapezoidal and rectangular channels. Proceed­ings of the Japan Society of Civil Engineers, No. 246, 1976-2, pp. 57-72 (in Japanese) (Transactions of the Japan Society of Civil Engineers, Vol. 8, Nov. 1977, pp. 122-125).

12. OHTSU, I. and YASUDA, Y., Discussion of "Hydraulic jump in triangular channel", IAHR, Journal of Hydraulic Research, Vol. 27. 1989, No. 1. pp. 178-188.

B. Discussor: LUDOVICO IVANFSSEVICH MACHADO, Consultant engineer, Buenos Aires, Argentine

This very interesting paper presents as a principal result the formula (10) which is only valid for an upstream bottom slope 8 = 45° and which not allows a direct solution. For example, knowing the discharge per unit width (q) and the sequent flow depth (h2) neither E nor /•", are known and therefore Y cannot be calculated. The writer proposes another solution. It can be accepted that the total head before the B-jump, referred to the horizontal bottom, is constant along the upstream bottom slope:

HT] = 11 + /;, + q2 ■ cos2 Ofig -h] = Hj (37)

Consequently although zx and h\ could be unknown, Hv would be easily calculated for a given discharge. Considering other adimensional variables, equation (2) can be written:

1 + 2fj2 + {<f>- \)(FX\F2 cos 0fB - 2f,4/3 ■ F22/3/cos4/3 0 = 0 (38)

where:

<t> = <f>(h2/H,) (39)

Taking into account the experimental data reproduced by the author in Appendix 1, and considering these adimensional variables, the writer has obtained by a conventional statistical analysis, the following formula with general good agreement:

F, = 1 . 4 / F 2 ° « . ( / ; 2 / / / T ) 1 ' 3 (40)

This equation always admits a direct solution. Analysing Peterka's experimental data of hydraulic jumps on mild slopes and Hager's tests, a universal formula is proposed:

/•j = 1 .4/^ . ( / , 2 / / / , ) b (41)

in which:

a = 0.3264/(tg $)°,566 and 6 = 1.296-(tg 6»)0174 (42)

This equation must be confirmed by experiments in slopes with angles comprised between 17° and 45°. For horizontal hydraulic jumps: «=0 .6195 ; b = 0.7273. Moreover, by statistical analysis a simple equation can be given to calculate the sequent depth in horizontal hydraulic jumps:

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/72(1 = 0.8565//ï2M,y4771 (43)

Thus, when an upstream slope exists, it can be stated that if: h2 > h2o then the jump will take place partially or totally on the upstream slope.

Reply by the author

A. Ohtsu and Yasuda's comments in their substantial contribution are welcomed as a valuable source of data. Given that the B-jump in sloping channels has received only limited attention, the writers con­siderably expand the original paper. In turn, the author has submitted a second paper on B-jumps meanwhile, where a channel with 6 = 30° is considered. The result ofthe paper under considera­tion have been generalised, particularly as regards the sequent depths ratio, and the length of roller. The writers develop a new expression for the sequent depths as given by equation (29). This rela­tion is quite involved and depends on the coefficients k' and /■. Apart from the fact that k' — 1 for tg 0 > 0.5(0 >26°), and r-» 1 for large tailwater submergence, no advantage of the writers' approach is seen. Ohtsu and Yasuda's final equation (36) depends on a large number of experi­mentally adjusted coefficients, whereas Fig. 10 is straightforward. Furthermore, the author's parameter E has some physical relevance as it describes the type of flow (Figs. 13 and 14). Regarding the length of'roller, Fig. 20 looks somewhat nicer than Fig. 10, but suffers from facts already discussed earlier [12]. Also, a straight curve somewhat steeper than Fig. 19 would fit the data better. The length of jump as defined by the writers lacks still a proper definition. The writers' classification of B-jumps as proposed in Table 3 is welcomed. Bl-jumps are partic­ularly interesting from the point of view of energy dissipation, and decay of maximum cross-sectional velocity. Typically, £-values of those jumps are in the domain 0.5 < E < 1. For B2-jumps the domain of E is 0.25 < E < 0.5. The surface roller may still clearly be noticed, but the flow resembles only partly to a jump. The air entrainment and turbulence level are reduced and the decay of maximum cross-sectional veloity is much smaller. For E < 0.25 one may no more talk of a hydraulic-jump. As will be seen from the paper already mentioned and presently under review, such flow is comparable to sloping wall jets. They cannot be recommended as energy dissipators, as is also noted from Fig. 8. B. Machado's discussion on the computation ofthe sequent flow depths is appreciated. The sequent depths h\ and h2 are depending on the discharge q per unit width and the toe eleva­tion z\ above the tailwater channel. The angle ofthe approaching channel is fixed and equal to 0 = 45°. Given that the approaching flow is supercritical and the spillway geometry known, the relation between discharge q and flow depth N\ or pressure head ht = A', cos 0 is known at each point of interest x. Asa result, h\(x) or also h\(z{) depend only on q. Therefore, the considerations include (q,huh2) or (q,zuh2) as independent parameters. In what follows the latter configuration of parameters will be referred to. Several possibilities are of engineering concern: 1. Given q and z,. What is the tailwater depth h2 needed? This question involves F\ as given

and both Y and E as unknowns. However, upon letting Zx=z\\hu one may express

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E = 1 — Z\ Y~', and equation (10) may implicitly be solved for Y = y(Z, f,). 2. Given q, h2 and sought is Z\. In this configuration, £ is the unknown and may implicitly be

solved from equation (10). 3. Given h2,z\ and sought q (and thus also A,). One must consider the function F\(Y) as unknown

and solve it for given E. The first case corresponds to the setting of the tailwater for given unit discharge and imposed toe positions. Such questions are rather seldom in the design procedure. The second case is more frequently encountered and involves the determination of the toe of jump for given inflow and tailwater conditions. The third case is rarely found in applications and involves discharges other than the design discharge. To come back to the discussor's case, the configuration of known unit discharge q and tailwater depth h2 yields an infinite number of solutions as the position of toe is not specified. However, if h\{z\) is known too, case 2 applies. The discussor treated a special case of constant energy head relative to the datum, a configuration which might approximate low head structures but cannot be related to high velocity inflow, as the frictional effects then become significant. Also, it should be noted that Machado's new param­eter (hijHj) contains besides {h2jh\) both Z, and F\, as

//■,■/*, = 1 + Z, + (1/2)F,2. (44)

At this point it is suggested that the solution of implicit equations is normally more satisfactory than the explicit result of an approximation whose degree of accuracy is unknown. Approxima­tions such as those of Machado or for uniform approaching flow may be used as a guess for the first iteration, of course.

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