Free surface flows over a successive obstacles with surface tension and gravity effects

Abdelkader Laiadi; Abdelkrim Merzougui; Abdelkader Laiadi; Abdelkrim Merzougui

doi:10.3934/math.2019.2.316

AIMS Mathematics

2019, Volume 4, Issue 2: 316-326. doi: 10.3934/math.2019.2.316

Previous Article Next Article

Research article

Free surface flows over a successive obstacles with surface tension and gravity effects

Abdelkader Laiadi ^{1
,
,},
Abdelkrim Merzougui ²

1.
Department of Mathematics, Biskra University, Biskra, Algeria
2.
Department of Mathematics, M’sila University, M’sila, Algeria

Received: 19 January 2019 Accepted: 26 March 2019 Published: 09 April 2019

The problem of steady two-dimensional flow of a fluid of finite depth over a successive obstacles is considered. Both gravity and surface tension are taken into account in the dynamic boundary conditions. The fluid is assumed to be inviscid, incompressible and the flow to be irrotational. The flow is characterized by the two parameters, the Froude number $Fr$ and the inverse Weber number $\delta$. The fully non-linear problem is solved numerically by using the boundary integral equation technique. The numerical solutions for sub-critical ($Fr < 1$) and supercritical ($Fr>1)$ are presented for various values of $Fr$ and $\delta$. The effects of surface tension and gravity on the shape of the free surface are discussed, and solution diagrams for all flow regimes are presented.
- free-surface flow,
- potential flow,
- Weber number,
- surface tension,
- Froude number,
- integro-differential equation
Citation: Abdelkader Laiadi, Abdelkrim Merzougui. Free surface flows over a successive obstacles with surface tension and gravity effects[J]. AIMS Mathematics, 2019, 4(2): 316-326. doi: 10.3934/math.2019.2.316

Related Papers:

Abstract

The problem of steady two-dimensional flow of a fluid of finite depth over a successive obstacles is considered. Both gravity and surface tension are taken into account in the dynamic boundary conditions. The fluid is assumed to be inviscid, incompressible and the flow to be irrotational. The flow is characterized by the two parameters, the Froude number $Fr$ and the inverse Weber number $\delta$. The fully non-linear problem is solved numerically by using the boundary integral equation technique. The numerical solutions for sub-critical ($Fr < 1$) and supercritical ($Fr>1)$ are presented for various values of $Fr$ and $\delta$. The effects of surface tension and gravity on the shape of the free surface are discussed, and solution diagrams for all flow regimes are presented.

References

[1]	M. B. Abd-el-Malek, S. N. Hanna and M. T. Kamel, Approximate solution of gravity flow from a uniform channel over triangular bottom for large Froude number, Appl. Math. Model., 15 (1991), 25-32. doi: 10.1016/0307-904X(91)90077-3
[2]	G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
[3]	S. R. Belward, Fully non-linear flow over successive obstacles: Hydraulic fall and supercritical flows, J. Austral. Math. Soc. Ser. B, 40 (1999), 447-458. doi: 10.1017/S0334270000010535
[4]	S. R. Belward and L. K. Forbes, Fully non-linear two-layer flow over arbitrary topography, J. Eng. Math., 27 (1993), 419-432. doi: 10.1007/BF00128764
[5]	B. J. Binder, Steady two-dimensional free surface flow past disturbances in an open channel: Solutions of the Korteweg-De Vries equation and analysis of the weakly nonlinear phase space, Fluids, 4 (2019), 24. Available from: https://www.mdpi.com/2311-5521/4/1/24.
[6]	B. J. Binder, M. G. Blyth and S. W. Mccue, Free-surface flow past arbitrary topography and an inverse approach for wave free solutions, IMA J. Appl. Math., 78 (2013), 685-696. doi: 10.1093/imamat/hxt015
[7]	B. J. Binder, F. Dias and J. M. Vanden-Broeck, Influence of rapid changes in a channel bottom on free surface flows, IMA J. Appl. Math., 73 (2008), 254-273.
[8]	B. J. Binder, F. Dias and J. M. Vanden-Broeck, Steady free surface flow past an uneven channel bottom, Theor. Comput. Fluid Dyn., 20 (2006), 125-144. doi: 10.1007/s00162-006-0017-y
[9]	B. J. Binder, J. M. Vanden-Broeck and F. Dias, Forced solitary waves and fronts past submerged obstacles, Chaos, 15 (2005), 037106.
[10]	F. Dias and J. M. Vanden-Broeck, Generalised critical free surface flows, J. Eng. Math., 42 (2002), 291-301. doi: 10.1023/A:1016111415763
[11]	F. Dias and J. M. Vanden-Broeck, Open channel flows with submerged obstructions, J. Fluid. Mech., 206 (1989), 155-170. doi: 10.1017/S0022112089002260
[12]	L. K. Forbes, Free-surface flow over a semicircular obstruction, including the influence of gravity and surface tension, J. Fluid. Mech., 127 (1983), 283-297. doi: 10.1017/S0022112083002724
[13]	L. K. Forbes and L. W. Schwartz, Free-surface flow over a semicircular obstruction, J. Fluid. Mech., 114 (1982), 299-314. doi: 10.1017/S0022112082000160
[14]	S. Grandison and J. M. Vanden-Broeck, Truncation approximations for gravity-capillary free surface flows, J. Eng. Math., 54 (2006), 89-97. doi: 10.1007/s10665-005-7719-9
[15]	P. Guayjarernpanishk and J. Asavanant, Free-surface flows over an obstacle: Problem revisited, In: Bock H., Hoang X., Rannacher R., Schlöder J. Editors, Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, Heidelberg, (2012), 139-151.
[16]	S. N. Hanna, Influence of surface tension on free surface flow over a polygonal and curved obstruction, J. Comput. Appl. Math., 51 (1994), 357-374. doi: 10.1016/0377-0427(92)00111-L
[17]	R. J. Holmes and G. C. Hoking, A note on waveless subcritical flow past symmetric bottom topography, Euro. J. Appl. Math., 28 (2016), 562-575.
[18]	R. J. Holmes, G. C. Hoking, L. K. Forbes, et al. Waveless subcritical flow past symmetric bottom topography, Euro. J. Appl. Math., 24 (2013), 213-230. doi: 10.1017/S0956792512000381
[19]	A. C. King and M. I. G. Bloor, Free-surface flow of a stream obstructed by an arbitrary bed topography, Q. J. Mech. Appl. Math., 43 (1990), 87-106. doi: 10.1093/qjmam/43.1.87
[20]	C. Lustri, S. W. Mccue and B. J. Binder, Free surface flow past topography: A beyond-all-orders approach, Euro. J. Appl. Math., 23 (2012), 441-467. doi: 10.1017/S0956792512000022
[21]	A. Merzougui and A. Laiadi, Free surface flow over a triangular depression, TWMS J. App. Eng. Math., 4 (2014), 67-73.
[22]	R. Pethiyagoda, T. J. Moroney and S. W. Mccue, Efficient computation of two-dimensional steady free surface flows, Int. J. Numer. Meth. Fluids, (2017), 1-20.
[23]	L. J. Pratt, On non-linear flow with multiple obstructions, J. Atmos. Sci., 41 (1984), 1214-1225. doi: 10.1175/1520-0469(1984)041<1214:ONFWMO>2.0.CO;2
[24]	J. M. Vanden-Broeck, Gravity-Capillary Free Surface Flows, New York: Cambridge University Press, 2010.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)