Roll waves and their generation criteria

de Oliveira Ferreira, Fabiana; Maciel, Geraldo de Freitas; Pereira, João Batista

doi:10.1590/2318-0331.262120200185

Fabiana de Oliveira Ferreira

Universidade Estadual Paulista, Ilha Solteira, SP, Brasil

http://orcid.org/0000-0001-7331-8587

Geraldo de Freitas Maciel

Universidade Estadual Paulista, Ilha Solteira, SP, Brasil

http://orcid.org/0000-0003-1272-9045

João Batista Pereira

Universidade Estadual Paulista, Ilha Solteira, SP, Brasil

http://orcid.org/0000-0001-7136-917X

E-mails: fabiana.of@gmail.com (FOF), geraldo.f.maciel@unesp.br (GFM), joao.pereira@unesp.br (JBP)

Authors contributions

Fabiana de Oliveira Ferreira: First author who contributed to literature review, Performed the simulation tests and discussion of results, as well as writing and formatting of the article.

Geraldo de Freitas Maciel: Research coordinator, paper conception, analysis and completion of results and discussion.

João Batista Pereira: Literature review, writing, revision and formatting of the article.

Editor Chefe: Adilson Pinheiro

Editor Associado: Iran Eduardo Lima Neto

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Figure 1
Roll wave phenomenon in open channels and in experimental tests carried out in laboratories. Roll waves in water: (a) Turner Dam and Reservoir, Vally Center (Ponce, 2005Ponce, V. M. (2005, 24 de fevereiro). Project Photos MM: Turner Reservoir, San Diego Country, California. Retrieved in 2020, October 27, from http://ponce.sdsu.edu/projectphotosmm.html
http://ponce.sdsu.edu/projectphotosmm.ht... ), (b) concrete channel, Lions Bay (Balmforth, 2011Balmforth, N. (2011). Roll waves in a concrete channel in Lions, Vancouver. Retrieved in 2020, October 15, from http://www.math.ubc.ca/~njb/Research/rollo.htm
http://www.math.ubc.ca/~njb/Research/rol... ), (e) Channelized Achumani River, La Paz (Ponce & Guzmán, 2019Ponce, V. M., & Guzmán, B. C. (2019, 13 de junho). The control of roll waves in channelized rivers. Retrieved in 2020, October 27, from http://ponce.sdsu.edu/the_control_of_roll_waves.html
http://ponce.sdsu.edu/the_control_of_rol... ). Roll waves in glycerin: (c) experimental apparatus for roll wave measurements, LH² Lab., Ilha Solteira, Brazil (own authorship). Roll waves in mud: (d) concrete channel (Institut National de Recherche en Sciences et Technologies pour L’environnement et L’agriculture, 2001Institut National de Recherche en Sciences et Technologies pour L’environnement et L’agriculture – IRSTEA. (2001). Connaitre les laves torrentielles. Grenoble: IRSTEA. Retrieved in 2020, October 27, from http://www.irstea.fr/lespace-jeunesse/nosressources/videos-0
http://www.irstea.fr/lespace-jeunesse/no... ), (f) experiment in an inclined channel, LH² Lab., Ilha Solteira, Brazil (own authorship).

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Figure 2
Flow curve of the fluids analyzed in the generation of experimental roll waves: (a) glycerin, (b) carbopol and (c) mud.

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Figure 2
Flow curve of the fluids analyzed in the generation of experimental roll waves: (a) glycerin, (b) carbopol and (c) mud.

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Figure 3
Numerical mesh used for roll wave simulation: (a) boundary conditions and (b) grid details.

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Figure 4
Numerical simulation versus mathematical model concerning the velocity profiles and the shear rates of the uniform flow for: (a) Newtonian, (b) Power-law and (c) Herschel-Bulkley fluids.

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Figure 5
Numerical simulations for the Newtonian fluid at different disturbance frequencies: 0.5, 1, 3. 4 and 6 Hz. Damping effect is observed when the frequency is higher than the cut-off frequency.

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Figure 6
Numerical simulations for non-Newtonian fluids at different disturbance frequencies: 0.8, 1.0, 2.0 and 3.7 Hz. Damping effect is observed when the frequency is higher than the cut-off frequency.

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Figure 7
Depths assessed on August 17^th, 1998 (Berti et al., 2000Berti, M., Genevois, R., LaHusen, R., Simoni, A., & Tecca, P. R. (2000). Debris flow monitoring in the Acquabona watershed on the Dolomites (Italian Alps). Physics and Chemistry of the Earth. Part B: Hydrology, Oceans and Atmosphere, 25(9), 707-715. http://dx.doi.org/10.1016/S1464-1909(00)00090-3.
http://dx.doi.org/10.1016/S1464-1909(00)... ). The highlighted wave train indicates a possible roll wave formation.

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Figure 8
Behavior of the cut-off frequency as a function of the Froude number for the different fluid rheologies evaluated in this work () and in the Acquabona case ().

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Figure A1
Behavior of the function

k_{i} {(ω_{r})}^{+}

(positive branch) and

k_{i} {(ω_{r})}^{-}

(negative branch), with

n = 0.4

,

C = 0.1

and varying Froude numbers.

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Figure A2
Behavior of the function

k_{r} {(ω_{r})}^{+}

(positive branch) and

k_{r} {(ω_{r})}^{-}

(negative branch), with

n = 0.4

,

C = 0.1

and varying Froude numbers.

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Figure A3
Influence exerted by the rheological parameters

n, C

on the cut-off frequency, when

F r > F r_{m i n}

: (a) Influence of parameter

C

, (b) Influence of the flow index of the fluid

n

.

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Table 1
Employed Numerical schemes in roll wave simulation.

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Table 2
Fluid properties, minimum conditions for roll wave generation, wave height

(Δ h),

wavelength

(λ)

and celerity

(U_{n u m}

).

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Table 3
Fluid properties and minimum conditions for roll wave generation in Acquabona case (Fiorot et al., 2018Fiorot, G. H., Ferreira, F. O., Dupont, P., & Maciel, G. F. (2018). Roll Waves mathematical model as a risk-Assessment tool: case study of Acquabona catchment. Journal of Hydraulic Engineering, 144(12), 05018009. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001538.
https://doi.org/10.1061/(ASCE)HY.1943-79... ).

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Fluid and Flow Properties	Newtonian			Power-law		Herschel-Bulkley
$τ_{c} (P a)$	0			0		0.20⁽¹ (1) Value attributed in the numerical test. ⁾
$K_{n} (P a . s^{n}) /$	0.212			0.14		0.14
$μ$ $(P a . s)$	0.212			0.14		0.14
$n$	1.00			0.40		0.40
$C$	0			0		0.1774
$α$	1.2			1.125		1.103
$h_{0} (m)$	0.0098			0.0057		0.0057
${\bar{u}}_{0} (m / s)$	0.256			0.235		0.123
$L_{m i n} (m)$ ⁽ ² (2) All minimal channel lengths Lmin (Di Cristo et al., 2013a) were verified. ⁾	0.45			1.26		1.35
$F r$	0.830			0.985		0.531
$F r_{m i n}$	0.577			0.298		0.218
$R e$	14.58			42.50		41.76
$f_{c} (H z)$	4.82			3.70		1.927
$ω_{c}$	1.163			0.563		0.558
Wave height and wavelength	Frequencies $(H z)$			Frequencies $(H z)$		Frequencies $(H z)$
	1.0	3.0	4.0	0.8	1.0	0.8	1.0
$ω_{c}$	0.241	0.723	0.965	0.121	0.152	0.232	0.291
$Δ h (m)$	0.0064	0.003	0.0011	0.0024	0.0021	0.0016	0.0009
$λ (m)$	0.7500	0.210	0.1500	0.6290	0.5030	0.5300	0.3500
$U_{n u m}$ (m/s)	0.750	0.630	0.600	0.5032	0.5030	0.424	0.350

Herschel-Bulkley model		Fluid and Flow properties
$τ_{c} (P a)$	$762$	$θ (d e g r e e s)$	7
$k_{n} (P a s^{n})$	458	$h_{0} (m)$	0.83
$n$	0.3	$α$	1.06
-	-	$ρ_{m i x t u r e} (k g / m^{3})$	2,039
-	-	$C$	0.377
-	-	${\bar{u}}_{0} (m / s)$	3.08
-	-	$F r$	1.08
-	-	$R e$	53.20
-	-	f (Hz)	$≅$ 0.01
Generation Criteria
$F r_{m i n}$	$f_{c} (H z)$	$ω_{c}$ (dimensionless cut-off frequency)
0.147	0.211	0.357

\frac{\partial (\bar{u} h)}{\partial t} + \frac{\partial (α {\bar{u}}^{2} h)}{\partial x} + \frac{1}{2 {Fr}^{2}} \frac{\partial h^{2}}{\partial x} = h - C - (1 - C) {[\bar{u} h \frac{(1 - C) (n + 1 + n C)}{(h - C) ((n + 1) h + n C)}]}^{n}

[1] E-mails: fabiana.of@gmail.com (FOF), geraldo.f.maciel@unesp.br (GFM), joao.pereira@unesp.br (JBP)

Multiphase model	VoF
Time	Unsteady
Time	Second Order Implicit
Viscous model	Laminar
Pressure-velocity coupling	Simplec
Pressure	Presto
Momentum	Third Order MUSCL
Volume fraction	Modified HRIC

Brasil

Brasil

Roll waves and their generation criteria

Roll waves e seus critérios de geração

ABSTRACT