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Hydraulic Jump in a Sloped Trapezoidal Channel
Available online at www.sciencedirect.com
ScienceDirect Energy Procedia 74 (2015) 251 – 257
International Conference on Technologies and Materials for Renewable Energy, Environment and Sustainability, TMREES15
Hydraulic jump in a sloped trapezoidal channel Samir kateba, Mahmoud Debabeche b , Ferhat Riguet a,b,* a
Assistant Professor, University of Ouargla.. Research Laboratory of Civil Engineering, Hydraulics, Sustainable Development and Environment (LARGHYDE), University of Biskra, Algeria. Corresponding author : [email protected] b Professor, Research Laboratory of Civil Engineering, Hydraulics, Sustainable Development and Environment (LARGHYDE), University of Biskra, PB 145 RP, 07000 Biskra, Algeria. PhD student, Laboratory of Civil Engineering, Hydraulics, Sustainable Development and Environment (LARGHYDE), University of Biskra, Algeria.
Abstract The hydraulic jump in a sloped trapezoidal channel is theoretically and experimentally examined. The study aims to determine the effect of the channel’s slope on the sequent depth ratio of the jump. A theoretical relation is proposed for the inflow Froude number as function of the sequent depth ratio and the channel slope. An experimental analysis is also proposed to find a better formulation of the obtained relation. For this motive, five positive slopes are tested. The relations obtained are recommended for designing irrigation ditches. Key words: Hydraulic jump, trapezoidal channel, positive slope, open channels, irrigation ditches.
Click insertPublished your abstract text. Ltd. This is an open access article under the CC BY-NC-ND license © 2015here Theand Authors. by Elsevier © 2015 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-reviewunder underresponsibility responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD). Peer-review of the Euro-Mediterranean Institute for Sustainable Development (EUMISD)
INTRODUCTION The hydraulic jump is used to dissipate the kinetic energy of a supercritical flow to avoid important modifications of the stilling basin bed. Nevertheless, the trapezoidal section does not satisfy the requirement of a stilling basin, but has some interesting practical applications when used as an irrigation ditch (Achour 1989). The capacity of the hydraulic jump to raise tailwater depth is used to prime a hose siphon designed for the required discharge. * Corresponding author. Tel.: +213-6672-18529; fax: +213-6672-18529.
E-mail address: [email protected]
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4 .0/). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) doi:10.1016/j.egypro.2015.07.591
252
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
The Hager and Wanoschek (1987), Achour and Debabeche (2003) and Debabeche and Achour (2007), examined the hydraulic jump evolving in a horizontal triangular channel. The first detailed study of the hydraulic jump, in a rectangular channel with positive slope, was by Bak hmeteff and Matzké (1938) who examined the surface profile, the length of the jump, and the velocity distribution. Kindsvater (1944) classified sloped jump according to the position of their toe with regard to the downstream extremity of the slope (Fig. 1), in four types: A-jump for which the toe of the jump coincides with the downstream extremity of the slope, B-jump for which the toe of the jump i s between the A-jump and the C-jump, C-jump for which the end of the jump roller coincides with the downstream extremity of the slope, and D-jump for which the jump roller appears completely in the sloped portion. The D-jump was analyzed by Wielogorski and Wilson (1970), Ohashi and al. (1 973), Rajaratnam and Muhrahari (1974), Mikhalev and Hoang (1976). The present study suggests investigating, theoretically and experimentally, the hydraulic jump in a trapezoidal channel and a positive slope. The configuration of the jump adopted for this paper corresponds to the D-jump according to the classification of Kindsvater (1944). The aim of this paper is to propose, for this configuration of jump, a theoretical relation ( F= f ( Y, λ , α) expressing the inflow Froude number F1 as a function of the angle of inclination α of the channel with regard to the horizontal of the sequent depth ratio ( Y = h2/h1)(h1and hare, inflow and final flow depths, respectively) and of the relative length (λ = Lj/ h1) o f the jump. The proposed relation will be obtained by application of the momentum equation applied between the upstream and downstream sections of the jump. In addition, an experimental analysis will be proposed to find a better formulation of the obtained theoretical relation. D C B A Cl
Fig. 1. Classification of sloped jumps according to Kindsvater (1944).
1. Theory The momentum equation applied between sections 1 and 2 is written as follows:
Fig. 2. Hydraulic jump in a sloped trapezoidal channel
ρQv1 P1 W sinα
ρQv 2 P2
(1)
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Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
where ρ is the density, and Q is flow discharge. Figure 3 shows a hydraulic jump evolving in a trapezoidal channel with positive slope. The following hypotheses will be considered in sections 1 and 2: that the pressure is hydrostatic and the friction forces are negligible. The weight W of the jump and the pressure forces P1 and P2 can be expressed by applying the hydrostatics laws as 2 h 2 cos D (2) 3b 2mh12 ; P2 Z h2 cos D 3b 2mh22 ; W ZV P1 Z 1 6
6
b+2mh1
b) 22
ħ1
1
h1
α b
ħ1
h1
Section 1-1 b+2mh2
ħ2 1
h2
ħ2 h2
α
α 2
b
2
a)
b)
Section 2-2
Fig. 3. a) Représentation géométrique du volume équivalent représentatif du ressaut. b) Représentation géométrique des sections amont et aval du ressaut.
Point gauge Trapézoïdal Channel Sill
h2
Lr
h1 Pressure boxes
Lj
Supply Pipeline PVC
Regulationg Vale Pump
Q
Tank
Electromagnetic Flowmetre Fig. 4. Experimental model
254
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
Where h1 and h2 are flow depths, v1 and v2 are average velocities, a is the angle of inclination of the channel, V is the volume of water included between sections 1 and 2; m is the cotangent of the angle of inclination q of the channel, the volume V of the jump in a trapezoidal channel bh1 mh12 bh2 mh22 L ; M mh1 ; Y h2 V j 2 b h1 V
bh1
1 M Y 1 YM u K u L 2
(3) j
By considering eq.(2) and eq.(3), eq.(1) can be written as
>
@
ª bh 1 M 2 º bh 2 1 M Y 1 YM º u k u 1 » 1 3 2M cos D bh 1 ª L J sin D F12 « « » « b 1 2M » 6 2 ¬ ¼ ¬ ¼
>
@
ª bhY 1 YM 2 º bY 2 h 2 1 1 » 3 2YM cos D F22 « « b 1 2YM » 6 ¬ ¼
(4)
Equation (4) may be expressed as follows: ª § 1 M ·º ¸¸» F12 «1 ¨¨ ¬ © Y 1 YM ¹¼
§ 1 2M ·§ ª § 2 1 ¸¨ Y 2 ¨1 YM ·¸ §¨1 2 M ·¸º 1 M Y 1 YM u O u k u tgα ·¸ cos D ¨ » ¸ ¨ 1 M 2 ¸¨© «¬ © 3 2 ¹ © 3 ¹¼ ¹ © ¹
(5)
Equation [6] expresses the inflow Froude number F1 as a function of the sequent depths ratio Y of the angle of inclination a of the channel and the relative length λ= Lj/h1 of a hydraulic jump with positive slope, evolving in α Trapezoidal channel. By putting α = 0 in eq. [5], one obtains eq. [6] of Hager and Wanoschek (1987) concerning the classical hydraulic Jump in a trapezoidal channel. ª 2M º 2YM 1 M º 1 2M ª 2 (6) ) ) (1 Y (1 F2 1 1
« ¬
» Y(1 MY) ¼
2(1 M)2 «¬
3
3 »¼
Experiments were made in trapezoidal channel a 10 m long and width 0,2 m of the channel; at the Research Laboratory in Subterranean and Surface Hydraulics (LARHYSS) of the hydraulic department of University of Biskra (Fig.5). The hydraulic jump was created by setting a sill at the channel extremity. Five positions of the slope were tested so that the tangent of the angle of inclination a with regard to the horizontal takes the following values: 0%, 0,5%,1%, 1,5%, 2%.For each value of the channel slope, a large range of the inflow Froude numbers was obtained (3,5 < F1 < 14,5).
a)
b) Fig. 5. photo of the experimental channel.
a) Profile view, b) Top view
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Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
a)
b) Fig. 6. a)Photo of a series of pressure boxes b)Photo of a series of sill
Fig. 7. Photo of the hydraulic jump evolving in a positive sloped trapezoidal channel F1 =8, 96; s =17 cm; Lj =164 cm; h2 =20,8 cm ;tang(D)=0,01
Fig. 8. Photo of the hydraulic jump evolving in a sloped null trapezoidal channel F1 =6,63; s =7 cm; Lj =110 cm; h2 =13,7 cm ; tang(D)=0
2. Determination of k coefficient From eq. (7) one obtains the following expression of the coefficient k: ª § 1 M ·º ª § 2 1 §¨ 1 2M ·¸ · § 2 ·º ¸¸» cos D «Y 2 ¨1 YM ¸ ¨1 M ¸» F12 «1 ¨¨ 2 ¸ ¨ 2 © 1 M ¹ ¹ © 3 ¹¼ (7) ¬ © 3 ¬ © Y 1 YM ¹¼ k 1 §¨ 1 2M ·¸ 1 M Y 1 YM u O u sinα 2 ¨© 1 M 2 ¸¹ The k value was found by regression by using experimental data and its value is k = 1,03. This coefficient is a constant and does not depend on the slope of the channel. McCorcodale and Mohamed (1994) also verified this observation as well as Pagliara and Peruginelli (2000) for the jump evolving in a rectangular channel with adverse slope. 1
F1 Th
§1 ·2 § · ¨ cos D ¨ 1 2M ¸§¨ ªY 2 §¨1 2 YM ·¸ §¨1 2 M ·¸º 1,30 1 M Y 1 YM u k u tgα ·¸ ¸ » ¸ ¨2 ¸ ¨ 1 M 2 ¸¨© «¬ © 3 ¹ © 3 ¹¼ ¹¸ © ¹ ¨ ¨ ¸ ª § 1 M ·º ¨ ¸ ¸¸» «1 ¨¨ Y 1 YM ¨ ¸ ¹¼ ¬ © © ¹
(8)
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Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257 12 Y 11 10 9 8 7 6 5 4 3 F1 2 1 1
2
3
4
5
6
7
8
9
10
11
12
Fig. 9. Variation of the sequent depth ratio Y as function of the inflow Froude number F1 for a sloped jump according to eq. (5), for Seven slopes (%): (□) 2, (×) 1.5, (Δ) 1, (ж) 0.5,( o) 0 (—) curve according to eq. (6). (- - - ) curves according to eq. (8).
3.1.Explicit relation of the sequent depths ratio Y(F1, i) One notices that eq. (5) appears under an implicit form with regard to the sequent depths ratio Y and its application Consequently requires the use of an iterative process. Figure 9 shows that for a fixed F1, the tail water depth h2 increases with an increase in the channel slope i. By using The experimental data, the analysis of eq. (5) finds the following equation of regression: (9) Y = (10,22 i + 1,18 ) F1 for i ≥ 0 i=tgα ≤ 0.02 and 1.5< F1 <12 Equation [9] gives a simple means for the determination of the sequent depths ratio Y, using the inflow Froude number F1 and the channel negative slope i. Conclusion: The hydraulic jump in a sloped trapezoidal channel with a is theoretically and experimentally studied. The configuration of the jump adopted for this paper corresponds to the D-jump type. A general relation is obtained for the inflow Froude number as function of the sequent depths ratio of the angle of inclination of the channel and of the relative length of the jump. The coefficient k, which represents the ratio between the weight of the real volume of the jump to the weight of the computed one, was found by using experimental data and it is a constant that does not depend on the inflow Froude number or the channel slope. However, the obtained relation appears under an implicit form with regard to the sequent depths ratio Y, and an explicit relation is proposed. This relation in particular permits a direct determination of the sequent depth ratio Y, knowing the inflow Froude number F1, and the slope of the channel.
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
References: [1] Achour, B. 1989. Jump flowmeter in a channel of triangular crosssection without weir. Journal of Hydraulic Research, 27(2): 205–214. [2] Achour, B., and Debabeche, M. 2003. Control of hydraulic jump by sill in a triangular channel. Journal of Hydraulic Research, 41(3): 97–103. [3] Bakhmeteff, B.A., and Matzke´, A.E. 1938. The hydraulic jump insloped channel. Transactions, ASME, 60(HYD-60-1): 111-118. [4] Hager, W.H. 1992. Energy dissipators and hydraulic jump. Kluwer Academic Publ., Water Science and Technology Library, Dordrecht, The Netherlands. Vol. 8, 288 p. [5] Hager, W.H., and Wanoschek, R. 1987. Hydraulic jump in triangular channel. Journal of Hydraulic Research, 25(5): 549–564. [6] Kindsvater, C.E. 1944. The hydraulic jump in sloping channels.Transactions ASCE, 109: 1107–1154. [7] Mikhalev, M.A., and Hoang, T.A. 1976. Kinematic caracetritics of a hydraulic jump on a sloping apron. Power Technology and Engineering,10(7): 686–690. [8] McCorcodale, J.A., and Mohamed, M.S. 1994. Hydraulic jumps on adverse slope. Journal of Hydraulic Research, 32(1): 119-130. [9] Ohashi, I., Sakabe, I., and Aki, S. 1973. Design of combined hydraulic jump and sky-jump energy dissipators of flood spillway.In Proceedings of the 13th Congress of ICOLD, Q.41, R.19.pp. 311–333. [10] Pagliara, S., and Peruginelli, A. 2000. Limiting and sill-controlled adverse-slope hydraulic jump. Journal of Hydraulic Engineering, 126(11): 847–850. doi:10.1061/(ASCE)0733-9429(2000) 126:11(847). [11] Rajaratnam, N., and Murahari, V. 1974. Flow characteristics of sloping channel jumps. Journal of Hydraulic Engineering, 100:731–740. [12] Wielogorski, W., and Wilson, E.H. 1970. Non-dimensional profile area coefficients for hydraulic jump in sloping rectangular channels. Water Power, 22(4): 144–150. [13] Kateb, S., Debabeche, M. and Benmalek, A. 2013. Étude expérimentale de l'effet de la marche positive sur le ressaut hydraulique évoluant dans un canal trapézoïdal Canadian Journal of Civil Engineering Rev.can.génieciv.40 :1014–1018(2013)
List of symbols: F1 inflow Froude number [–] g acceleration due to gravity [m_s–2] h1 upstream sequent depth [m] h2 downstream sequent depth [m] i channel slope (i = tg(α)) Lj length of jump [m] k coefficient of correction of the jump weight [–] m cotangent of the angle of inclination of the channel walls with regard to the horizontal [–] Q flow discharge [m 3_s–1] V volume of the jump included between sections 1and 2 [m3] Y sequent depth ratio (Y = h2/h1) [–] α angle of inclination of the channel with regard to the horizontal [rad] λ relative length of jump (λ = Lj/h1) [–]
257
ScienceDirect Energy Procedia 74 (2015) 251 – 257
International Conference on Technologies and Materials for Renewable Energy, Environment and Sustainability, TMREES15
Hydraulic jump in a sloped trapezoidal channel Samir kateba, Mahmoud Debabeche b , Ferhat Riguet a,b,* a
Assistant Professor, University of Ouargla.. Research Laboratory of Civil Engineering, Hydraulics, Sustainable Development and Environment (LARGHYDE), University of Biskra, Algeria. Corresponding author : [email protected] b Professor, Research Laboratory of Civil Engineering, Hydraulics, Sustainable Development and Environment (LARGHYDE), University of Biskra, PB 145 RP, 07000 Biskra, Algeria. PhD student, Laboratory of Civil Engineering, Hydraulics, Sustainable Development and Environment (LARGHYDE), University of Biskra, Algeria.
Abstract The hydraulic jump in a sloped trapezoidal channel is theoretically and experimentally examined. The study aims to determine the effect of the channel’s slope on the sequent depth ratio of the jump. A theoretical relation is proposed for the inflow Froude number as function of the sequent depth ratio and the channel slope. An experimental analysis is also proposed to find a better formulation of the obtained relation. For this motive, five positive slopes are tested. The relations obtained are recommended for designing irrigation ditches. Key words: Hydraulic jump, trapezoidal channel, positive slope, open channels, irrigation ditches.
Click insertPublished your abstract text. Ltd. This is an open access article under the CC BY-NC-ND license © 2015here Theand Authors. by Elsevier © 2015 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-reviewunder underresponsibility responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD). Peer-review of the Euro-Mediterranean Institute for Sustainable Development (EUMISD)
INTRODUCTION The hydraulic jump is used to dissipate the kinetic energy of a supercritical flow to avoid important modifications of the stilling basin bed. Nevertheless, the trapezoidal section does not satisfy the requirement of a stilling basin, but has some interesting practical applications when used as an irrigation ditch (Achour 1989). The capacity of the hydraulic jump to raise tailwater depth is used to prime a hose siphon designed for the required discharge. * Corresponding author. Tel.: +213-6672-18529; fax: +213-6672-18529.
E-mail address: [email protected]
1876-6102 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4 .0/). Peer-review under responsibility of the Euro-Mediterranean Institute for Sustainable Development (EUMISD) doi:10.1016/j.egypro.2015.07.591
252
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
The Hager and Wanoschek (1987), Achour and Debabeche (2003) and Debabeche and Achour (2007), examined the hydraulic jump evolving in a horizontal triangular channel. The first detailed study of the hydraulic jump, in a rectangular channel with positive slope, was by Bak hmeteff and Matzké (1938) who examined the surface profile, the length of the jump, and the velocity distribution. Kindsvater (1944) classified sloped jump according to the position of their toe with regard to the downstream extremity of the slope (Fig. 1), in four types: A-jump for which the toe of the jump coincides with the downstream extremity of the slope, B-jump for which the toe of the jump i s between the A-jump and the C-jump, C-jump for which the end of the jump roller coincides with the downstream extremity of the slope, and D-jump for which the jump roller appears completely in the sloped portion. The D-jump was analyzed by Wielogorski and Wilson (1970), Ohashi and al. (1 973), Rajaratnam and Muhrahari (1974), Mikhalev and Hoang (1976). The present study suggests investigating, theoretically and experimentally, the hydraulic jump in a trapezoidal channel and a positive slope. The configuration of the jump adopted for this paper corresponds to the D-jump according to the classification of Kindsvater (1944). The aim of this paper is to propose, for this configuration of jump, a theoretical relation ( F= f ( Y, λ , α) expressing the inflow Froude number F1 as a function of the angle of inclination α of the channel with regard to the horizontal of the sequent depth ratio ( Y = h2/h1)(h1and hare, inflow and final flow depths, respectively) and of the relative length (λ = Lj/ h1) o f the jump. The proposed relation will be obtained by application of the momentum equation applied between the upstream and downstream sections of the jump. In addition, an experimental analysis will be proposed to find a better formulation of the obtained theoretical relation. D C B A Cl
Fig. 1. Classification of sloped jumps according to Kindsvater (1944).
1. Theory The momentum equation applied between sections 1 and 2 is written as follows:
Fig. 2. Hydraulic jump in a sloped trapezoidal channel
ρQv1 P1 W sinα
ρQv 2 P2
(1)
253
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
where ρ is the density, and Q is flow discharge. Figure 3 shows a hydraulic jump evolving in a trapezoidal channel with positive slope. The following hypotheses will be considered in sections 1 and 2: that the pressure is hydrostatic and the friction forces are negligible. The weight W of the jump and the pressure forces P1 and P2 can be expressed by applying the hydrostatics laws as 2 h 2 cos D (2) 3b 2mh12 ; P2 Z h2 cos D 3b 2mh22 ; W ZV P1 Z 1 6
6
b+2mh1
b) 22
ħ1
1
h1
α b
ħ1
h1
Section 1-1 b+2mh2
ħ2 1
h2
ħ2 h2
α
α 2
b
2
a)
b)
Section 2-2
Fig. 3. a) Représentation géométrique du volume équivalent représentatif du ressaut. b) Représentation géométrique des sections amont et aval du ressaut.
Point gauge Trapézoïdal Channel Sill
h2
Lr
h1 Pressure boxes
Lj
Supply Pipeline PVC
Regulationg Vale Pump
Q
Tank
Electromagnetic Flowmetre Fig. 4. Experimental model
254
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
Where h1 and h2 are flow depths, v1 and v2 are average velocities, a is the angle of inclination of the channel, V is the volume of water included between sections 1 and 2; m is the cotangent of the angle of inclination q of the channel, the volume V of the jump in a trapezoidal channel bh1 mh12 bh2 mh22 L ; M mh1 ; Y h2 V j 2 b h1 V
bh1
1 M Y 1 YM u K u L 2
(3) j
By considering eq.(2) and eq.(3), eq.(1) can be written as
>
@
ª bh 1 M 2 º bh 2 1 M Y 1 YM º u k u 1 » 1 3 2M cos D bh 1 ª L J sin D F12 « « » « b 1 2M » 6 2 ¬ ¼ ¬ ¼
>
@
ª bhY 1 YM 2 º bY 2 h 2 1 1 » 3 2YM cos D F22 « « b 1 2YM » 6 ¬ ¼
(4)
Equation (4) may be expressed as follows: ª § 1 M ·º ¸¸» F12 «1 ¨¨ ¬ © Y 1 YM ¹¼
§ 1 2M ·§ ª § 2 1 ¸¨ Y 2 ¨1 YM ·¸ §¨1 2 M ·¸º 1 M Y 1 YM u O u k u tgα ·¸ cos D ¨ » ¸ ¨ 1 M 2 ¸¨© «¬ © 3 2 ¹ © 3 ¹¼ ¹ © ¹
(5)
Equation [6] expresses the inflow Froude number F1 as a function of the sequent depths ratio Y of the angle of inclination a of the channel and the relative length λ= Lj/h1 of a hydraulic jump with positive slope, evolving in α Trapezoidal channel. By putting α = 0 in eq. [5], one obtains eq. [6] of Hager and Wanoschek (1987) concerning the classical hydraulic Jump in a trapezoidal channel. ª 2M º 2YM 1 M º 1 2M ª 2 (6) ) ) (1 Y (1 F2 1 1
« ¬
» Y(1 MY) ¼
2(1 M)2 «¬
3
3 »¼
Experiments were made in trapezoidal channel a 10 m long and width 0,2 m of the channel; at the Research Laboratory in Subterranean and Surface Hydraulics (LARHYSS) of the hydraulic department of University of Biskra (Fig.5). The hydraulic jump was created by setting a sill at the channel extremity. Five positions of the slope were tested so that the tangent of the angle of inclination a with regard to the horizontal takes the following values: 0%, 0,5%,1%, 1,5%, 2%.For each value of the channel slope, a large range of the inflow Froude numbers was obtained (3,5 < F1 < 14,5).
a)
b) Fig. 5. photo of the experimental channel.
a) Profile view, b) Top view
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Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
a)
b) Fig. 6. a)Photo of a series of pressure boxes b)Photo of a series of sill
Fig. 7. Photo of the hydraulic jump evolving in a positive sloped trapezoidal channel F1 =8, 96; s =17 cm; Lj =164 cm; h2 =20,8 cm ;tang(D)=0,01
Fig. 8. Photo of the hydraulic jump evolving in a sloped null trapezoidal channel F1 =6,63; s =7 cm; Lj =110 cm; h2 =13,7 cm ; tang(D)=0
2. Determination of k coefficient From eq. (7) one obtains the following expression of the coefficient k: ª § 1 M ·º ª § 2 1 §¨ 1 2M ·¸ · § 2 ·º ¸¸» cos D «Y 2 ¨1 YM ¸ ¨1 M ¸» F12 «1 ¨¨ 2 ¸ ¨ 2 © 1 M ¹ ¹ © 3 ¹¼ (7) ¬ © 3 ¬ © Y 1 YM ¹¼ k 1 §¨ 1 2M ·¸ 1 M Y 1 YM u O u sinα 2 ¨© 1 M 2 ¸¹ The k value was found by regression by using experimental data and its value is k = 1,03. This coefficient is a constant and does not depend on the slope of the channel. McCorcodale and Mohamed (1994) also verified this observation as well as Pagliara and Peruginelli (2000) for the jump evolving in a rectangular channel with adverse slope. 1
F1 Th
§1 ·2 § · ¨ cos D ¨ 1 2M ¸§¨ ªY 2 §¨1 2 YM ·¸ §¨1 2 M ·¸º 1,30 1 M Y 1 YM u k u tgα ·¸ ¸ » ¸ ¨2 ¸ ¨ 1 M 2 ¸¨© «¬ © 3 ¹ © 3 ¹¼ ¹¸ © ¹ ¨ ¨ ¸ ª § 1 M ·º ¨ ¸ ¸¸» «1 ¨¨ Y 1 YM ¨ ¸ ¹¼ ¬ © © ¹
(8)
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Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257 12 Y 11 10 9 8 7 6 5 4 3 F1 2 1 1
2
3
4
5
6
7
8
9
10
11
12
Fig. 9. Variation of the sequent depth ratio Y as function of the inflow Froude number F1 for a sloped jump according to eq. (5), for Seven slopes (%): (□) 2, (×) 1.5, (Δ) 1, (ж) 0.5,( o) 0 (—) curve according to eq. (6). (- - - ) curves according to eq. (8).
3.1.Explicit relation of the sequent depths ratio Y(F1, i) One notices that eq. (5) appears under an implicit form with regard to the sequent depths ratio Y and its application Consequently requires the use of an iterative process. Figure 9 shows that for a fixed F1, the tail water depth h2 increases with an increase in the channel slope i. By using The experimental data, the analysis of eq. (5) finds the following equation of regression: (9) Y = (10,22 i + 1,18 ) F1 for i ≥ 0 i=tgα ≤ 0.02 and 1.5< F1 <12 Equation [9] gives a simple means for the determination of the sequent depths ratio Y, using the inflow Froude number F1 and the channel negative slope i. Conclusion: The hydraulic jump in a sloped trapezoidal channel with a is theoretically and experimentally studied. The configuration of the jump adopted for this paper corresponds to the D-jump type. A general relation is obtained for the inflow Froude number as function of the sequent depths ratio of the angle of inclination of the channel and of the relative length of the jump. The coefficient k, which represents the ratio between the weight of the real volume of the jump to the weight of the computed one, was found by using experimental data and it is a constant that does not depend on the inflow Froude number or the channel slope. However, the obtained relation appears under an implicit form with regard to the sequent depths ratio Y, and an explicit relation is proposed. This relation in particular permits a direct determination of the sequent depth ratio Y, knowing the inflow Froude number F1, and the slope of the channel.
Samir kateb et al. / Energy Procedia 74 (2015) 251 – 257
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List of symbols: F1 inflow Froude number [–] g acceleration due to gravity [m_s–2] h1 upstream sequent depth [m] h2 downstream sequent depth [m] i channel slope (i = tg(α)) Lj length of jump [m] k coefficient of correction of the jump weight [–] m cotangent of the angle of inclination of the channel walls with regard to the horizontal [–] Q flow discharge [m 3_s–1] V volume of the jump included between sections 1and 2 [m3] Y sequent depth ratio (Y = h2/h1) [–] α angle of inclination of the channel with regard to the horizontal [rad] λ relative length of jump (λ = Lj/h1) [–]
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