Wave length and celerity downstream from a hydraulic jump

Gomes, Ana Paula; Marzec, Eduardo Pivatto; Endres, Luiz Augusto Magalhães

doi:10.1590/2318-0331.241920180015

Ana Paula Gomes

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

http://orcid.org/0000-0002-2139-7639

Eduardo Pivatto Marzec

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

Luiz Augusto Magalhães Endres

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

http://orcid.org/0000-0003-4228-3187

E-mails: anapaulaa.gomes@gmail.com (APG), eduardomarzec@gmail.com@gmail.com (EPM), endres@ufrgs.br (LAME)

Authors contributions Ana Paula Gomes: Lead author of the paper, she participated in all phases of the study.

Eduardo Pivatto Marzec: participated in the study and his specific contribution was: design and research design, data acquisition, data analysis and Interpretation.

Luiz Augusto Magalhães Endres: Writing of the manuscript and critical revision of the manuscript regarding the intellectual content.

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Figure 1
Hydraulic jump formed downstream from a floodgate (a) and downstream from a spillway (b). Source: Adapted from Trierweiler Neto (2006)TRIERWEILER NETO, E.F. Avaliação do campo de pressões em ressalto hidráulico formado a jusante de uma comporta com diferentes graus de submergência. 2006. 174 f. Dissertação (Mestrado em Engenharia) - Universidade Federal do Rio Grande do Sul, Porto Alegre, 2006..

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Figure 2
Characteristic frequencies of level oscillation downstream from a hydraulic jump. Source: Adapted from Mok (2004)MOK, K. M. Relation of surface roller eddy formation and surface fluctuation in hydraulic jumps. Journal of Hydraulic Research, v. 42, n. 2, p. 207-212, 2004. http://dx.doi.org/10.1080/00221686.2004.9728383.
http://dx.doi.org/10.1080/00221686.2004.... .

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Figure 3
Types of vertical current profiles described as (from left to right): (I) uniform; (II) with constant vorticity; (III) exponential; and (IV) cosine. Source: Neves and Dias (2013)NEVES, C.; DIAS, A. A importância da caracterização da corrente para a correta determinação do comprimento e da altura das ondas. Parte 1: embasamento teórico e dispersão. Revista Brasileira de Recursos Hídricos, v. 18, n. 3, p. 215-229, 2013. http://dx.doi.org/10.21168/rbrh.v18n3.p215-229.
http://dx.doi.org/10.21168/rbrh.v18n3.p2... .

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Figure 4
Schematic representation of the model used (flow direction from right to left).

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Figure 5
Position of the ADV and orientation of the positive velocity components measured by the ADV in the x- (Vx), y- (Vy) and z- (Vz) directions.

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Figure 6
Characteristic wave periods for free and submerged hydraulic jump conditions (mean values derived from experiments). Tp, Ts, and Tmean are the peak, significant, and mean periods, respectively.

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Figure 7
Comparison between the frequencies derived from the free surface variation downstream from the free hydraulic jump and the dimensionless theoretical frequency of vortex formation (Strouhal number).

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Figure 8
Mean measured velocity components, (a) Vx, (b) Vy, and (c) Vz, considering the free hydraulic jump condition.

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Figure 9
Mean measured velocity components, (a) Vx, (b) Vy, and (c) Vz, considering the submerged hydraulic jump condition with 10% submergence.

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Figure 10
Mean measured velocity components, (a) Vx, (b) Vy, and (c) Vz, considering the submerged hydraulic jump condition with 20% submergence.

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Figure 11
Wave length for each characteristic wave period and for the free and submerged hydraulic jump conditions; Lp, Ls, and Lmean are the wave lengths associated with the peak, significant, and mean periods, respectively.

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Figure 12
Wave celerity for each characteristic wave period and hydraulic jump condition (free and submerged). Cobs is the observed wave celerity and Cp, Cs, and Cmean are the theoretical wave celerities associated with the peak, significant, and mean periods, respectively.

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Table 1
Equations for the determination of the roller length as suggested by several authors.

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Table 2
Equations for the determination of the free and submerged hydraulic jump length as suggested by several authors.

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Table 3
Flow parameters and positions of the measuring level probes in the performed experiments for free and submerged hydraulic jumps.

(1)

(2)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Author	Proposed Equation
Douma (1942, apud Teixeira, 2003TEIXEIRA, E. D. Previsão dos valores de pressão junto ao fundo em bacias de dissipação por ressalto hidráulico. 2003. 113 f. Dissertação (Mestrado em Engenharia) - Programa de Pós-graduação em Recursos Hídricos e Saneamento Ambiental, Instituto de Pesquisas Hidráulicas, Universidade Federal do Rio Grande do Sul, Porto Alegre, 2003.)	$L_{r} = 3 Y_{2}$	Equation 3
Peterka (1974)PETERKA, A. J. Hydraulic desing of stilling basins and energy dissipators. Denver: U.S. Department Interior, Bureau of Reclamation, 1974. (Engeneering Monograph, 25).	$L_{r} = 4.5 Y_{2.} (4.5 \leq F r_{1} \leq 9)$	Equation 4
Newmham (1973 apud Teixeira, 2003TEIXEIRA, E. D. Previsão dos valores de pressão junto ao fundo em bacias de dissipação por ressalto hidráulico. 2003. 113 f. Dissertação (Mestrado em Engenharia) - Programa de Pós-graduação em Recursos Hídricos e Saneamento Ambiental, Instituto de Pesquisas Hidráulicas, Universidade Federal do Rio Grande do Sul, Porto Alegre, 2003.)	$L_{r} = 6.3 (F r_{1} - 1)$	Equation 5
Marques et al. (1997)MARQUES, M. G.; DRAPEAU, J.; VERRETTE, J. L. Flutuação de pressão em um ressalto hidráulico. In: CONGRESSO LATINOAMERICANO DE HIDRÁULICA, 17., 1997, Guayaquil. Anais… Equador, 1997.	$L_{r} = 6 (Y_{2} - Y_{1})$	Equation 6

Author	Proposed Equation
Elevatorski (1959)ELEVATORSKI, A. E. Hydraulic energy dissipators. New York: McGraw-Hill, 1959.	$L_{j} = 6.9 (Y_{2} - Y_{1})$	Equation 7
Marques, Drapeau and Verrette (1997)MARQUES, M. G.; DRAPEAU, J.; VERRETTE, J. L. Flutuação de pressão em um ressalto hidráulico. In: CONGRESSO LATINOAMERICANO DE HIDRÁULICA, 17., 1997, Guayaquil. Anais… Equador, 1997.	$L_{j} = 8.5 (Y_{2} - Y_{1})$	Equation 8
Teixeira (2003)TEIXEIRA, E. D. Previsão dos valores de pressão junto ao fundo em bacias de dissipação por ressalto hidráulico. 2003. 113 f. Dissertação (Mestrado em Engenharia) - Programa de Pós-graduação em Recursos Hídricos e Saneamento Ambiental, Instituto de Pesquisas Hidráulicas, Universidade Federal do Rio Grande do Sul, Porto Alegre, 2003.	$L_{j} = 8. (Y_{2} - Y_{1})$	Equation 9
Lopardo et al. (2004)LOPARDO, R. A.; FATTOR, C. A.; CASADO, J. M.; LOPARDO, M. C. Uma aproximación turbulencia remanente de la longitud del Resalto Sumergido. In: CONGRESSO LATINO AMERICANO DE HIDRÁULICA, 21., 2004, São Pedro. Anais... São Pedro: International Association of Hydro-Environment Engineering and Research, 2004. 9 p.	$L_{s j} / (T_{w} - Y_{3}) = 41 (2 S ) + 6$ , where $S * = (T_{w} - Y_{2}) / Y_{2}$*	Equation 10

Q (l/s)	qp (m³/s/m)	Fr1	Probe	S=1.0: d2=265mm	S=1.1: T_w=289mm	S=1.2: T_w=317mm
40	0.10	7.4	1st	3.05	3.21	3.3
40	0.10	7.4	2nd	4.05	4.21	4.3
Q (l/s)	qp (m³/s/m)	Fr1	Probe	S=1.0: d2=339mm	S=1.1: T_w=375mm	S=1.2: T_w=408mm
60	0.15	7.0	1st	4.17	4.12	4.29
60	0.15	7.0	2nd	5.17	5.12	5.29
Q (l/s)	qp (m³/s/m)	Fr1	Probe	S=1.0: d2=409mm	S=1.1: T_w=449mm	S=1.2: T_w=490mm
80	0.20	6.9	1st	4.76	4.78	4.74
80	0.20	6.9	2nd	5.76	5.78	5.74

{(ω - k U_{s})}^{2} = \{\begin{matrix} \frac{g k^{2} h \tan β h}{β h + γ h t a n γ h t a n β h}, s e k^{2} < γ^{2} \\ \frac{g k^{2} h}{1 + γ h t a n γ h}, s e k^{2} = γ^{2} \\ \frac{g k^{2} h t a n h β h}{β h + γ h t a n γ h t a n h β h}, s e k^{2} > γ^{2} \end{matrix}

[1] E-mails: anapaulaa.gomes@gmail.com (APG), eduardomarzec@gmail.com@gmail.com (EPM), endres@ufrgs.br (LAME)

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Wave length and celerity downstream from a hydraulic jump

Comprimento e celeridade das ondas a jusante de ressalto hidráulico

ABSTRACT