The influence of surface tension and gravity on cavitating flow past an inclined plate in a channel
Author:
Abdelkader Laiadi
Journal:
Quart. Appl. Math. 80 (2022), 529-548
MSC (2020):
Primary 76B07, 76B10, 65M38; Secondary 35Q35
DOI:
https://doi.org/10.1090/qam/1617
Published electronically:
March 21, 2022
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper concerns the two-dimensional free surface cavity flow past an inclined plate in a finite depth. To close the cavity, a Riabouchinsky model is considered. The fluid is assumed to be inviscid and incompressible and the flow to be steady and irrotational. When the gravity and surface tension are negligible, an exact free streamline solution is derived. We solve the cavity flow problem by using two numerical methods. These methods allow us to compute solutions including the effects of gravity and surface tension. The first method is the series truncation and the second is the boundary integral equations, based on Cauchy integral formula. Numerical solutions are found for different values of the angle of inclination $\gamma$ and for various values of the Weber number, the Froude number. Good agreement between the two numerical schemes and the exact solution provides a check on the numerical methods.
References
- Alex Doak and Jean-Marc Vanden-Broeck, Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge, J. Engrg. Math. 126 (2021), Paper No. 8, 19. MR 4206636, DOI 10.1007/s10665-020-10086-z
- C. D. Andersson and J.-M. Vanden-Broeck, Bow flows with surface tension, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1952, 1985–1997. MR 1421737, DOI 10.1098/rspa.1996.0105
- Garrett Birkhoff and E. H. Zarantonello, Jets, wakes, and cavities, Academic Press, Inc., Publishers, New York, 1957. MR 0088230
- 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.
- B. J. Binder, J.-M. Vanden-Broeck, and F. Dias, Forced solitary waves and fronts past submerged obstacles, Chaos 15 (2005), no. 3, 037106, 13. MR 2184882, DOI 10.1063/1.1992407
- Charles W. Lenau and Robert L. Street, A non-linear theory for symmetric, supercavitating flow in a gravity field, J. Fluid Mech. 21 (1965), 257–280. MR 173432, DOI 10.1017/S0022112065000174
- Frédéric Dias and Jean-Marc Vanden-Broeck, Open channel flows with submerged obstructions, J. Fluid Mech. 206 (1989), 155–170. MR 1016912, DOI 10.1017/S0022112089002260
- D. Daboussy, F. Dias, and J.-M. Vanden-Broeck, Gravity flows with a free surface of finite extent, Eur. J. Mech. B Fluids 17 (1998), no. 1, 19–31. MR 1614533, DOI 10.1016/S0997-7546(98)80050-2
- D. Daboussy, F. Dias, and J.-M. Vanden-Broeck, On explicit solutions of the free-surface Euler equations in the presence of gravity, Phys. Fluids 9 (1997), no. 10, 2828–2834. MR 1472436, DOI 10.1063/1.869395
- S. Fluge, Fluid Dynamics, Spinger, Berlin, 1960.
- M. I. Gurevich, Theory of Jets in Ideal Fluids, Academic Press, New York and London, 1965.
- S. N. Hanna, M. N. Abdel-Malek, and M. B. Abd-el-Malek, Super-critical free-surface flow over a trapezoidal obstacle, Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), 1996, pp. 279–291. MR 1393736, DOI 10.1016/0377-0427(95)00160-3
- Jongwoo Lee and Jean-Marc Vanden-Broeck, Two-dimensional jets falling from funnels and nozzles, Phys. Fluids A 5 (1993), no. 10, 2454–2460. MR 1240907, DOI 10.1063/1.858758
- Abdelkader Laiadi and Abdelkrim Merzougui, Numerical solution of a cavity problem under surface tension effect, Math. Methods Appl. Sci. 44 (2021), no. 10, 8463–8471. MR 4271087, DOI 10.1002/mma.6474
- Abdelkader Laiadi and Abdelkrim Merzougui, Free surface flows over a successive obstacles with surface tension and gravity effects, AIMS Math. 4 (2019), no. 2, 316–326. MR 4135090, DOI 10.3934/math.2019.2.316
- L. M. Milne-Thomson, Theoretical hydrodynamics, Macmillan and Co. Ltd, 1962.
- D. Riabouchinsky, On steady fluid motions with free surfaces, Proc. London Math. Sot. II. 19 (1921), 206–215.
- Ravindra Pethiyagoda, Timothy J. Moroney, and Scott W. McCue, Efficient computation of two-dimensional steady free-surface flows, Internat. J. Numer. Methods Fluids 86 (2018), no. 9, 607–624. MR 3767309, DOI 10.1002/fld.4469
- J. M. Vanden-Broeck and F. Dias, Free-surface flows with two stagnation points, J. Fluid Mech. 324 (1996), 393–406.
- J. M. Vanden-Broeck, The influence of capillarity on cavitating flow past a flat plate, Q. J. Mech. Appl. Math. 34 (1981), 465–473.
- Jean-Marc Vanden-Broeck, Gravity-capillary free-surface flows, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 2010. MR 2722683, DOI 10.1017/CBO9780511730276
- J.-M. Vanden-Broeck, Nonlinear capillary free-surface flows, J. Engrg. Math. 50 (2004), no. 4, 415–426. MR 2116762, DOI 10.1007/s10665-004-1769-2
- J. M. Vanden-Broeck, Cavitating flow of a fluid with surface tension past a circular cylinder, Phys. Fluids. A. 3 (1991), no. 2, 263–266.
- L. C. Wrobel, A simple and efficient BEM algorithm for planar cavity flows, Int. J. Numer. Meth. Fluids. 14 (1992), no. 5, 529–537.
References
- Alex Doak and Jean-Marc Vanden-Broeck, Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge, J. Engrg. Math. 126 (2021), Paper No. 8, 19. MR 4206636, DOI 10.1007/s10665-020-10086-z
- C. D. Andersson and J.-M. Vanden-Broeck, Bow flows with surface tension, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1952, 1985–1997. MR 1421737, DOI 10.1098/rspa.1996.0105
- Garrett Birkhoff and E. H. Zarantonello, Jets, wakes, and cavities, Academic Press, Inc., Publishers, New York, 1957. MR 0088230
- 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.
- B. J. Binder, J.-M. Vanden-Broeck, and F. Dias, Forced solitary waves and fronts past submerged obstacles, Chaos 15 (2005), no. 3, 037106, 13. MR 2184882, DOI 10.1063/1.1992407
- Charles W. Lenau and Robert L. Street, A non-linear theory for symmetric, supercavitating flow in a gravity field, J. Fluid Mech. 21 (1965), 257–280. MR 173432, DOI 10.1017/S0022112065000174
- Frédéric Dias and Jean-Marc Vanden-Broeck, Open channel flows with submerged obstructions, J. Fluid Mech. 206 (1989), 155–170. MR 1016912, DOI 10.1017/S0022112089002260
- D. Daboussy, F. Dias, and J.-M. Vanden-Broeck, Gravity flows with a free surface of finite extent, Eur. J. Mech. B Fluids 17 (1998), no. 1, 19–31. MR 1614533, DOI 10.1016/S0997-7546(98)80050-2
- D. Daboussy, F. Dias, and J.-M. Vanden-Broeck, On explicit solutions of the free-surface Euler equations in the presence of gravity, Phys. Fluids 9 (1997), no. 10, 2828–2834. MR 1472436, DOI 10.1063/1.869395
- S. Fluge, Fluid Dynamics, Spinger, Berlin, 1960.
- M. I. Gurevich, Theory of Jets in Ideal Fluids, Academic Press, New York and London, 1965.
- S. N. Hanna, M. N. Abdel-Malek, and M. B. Abd-el-Malek, Super-critical free-surface flow over a trapezoidal obstacle, Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), 1996, pp. 279–291. MR 1393736, DOI 10.1016/0377-0427(95)00160-3
- Jongwoo Lee and Jean-Marc Vanden-Broeck, Two-dimensional jets falling from funnels and nozzles, Phys. Fluids A 5 (1993), no. 10, 2454–2460. MR 1240907, DOI 10.1063/1.858758
- Abdelkader Laiadi and Abdelkrim Merzougui, Numerical solution of a cavity problem under surface tension effect, Math. Methods Appl. Sci. 44 (2021), no. 10, 8463–8471. MR 4271087, DOI 10.1002/mma.6474
- Abdelkader Laiadi and Abdelkrim Merzougui, Free surface flows over a successive obstacles with surface tension and gravity effects, AIMS Math. 4 (2019), no. 2, 316–326. MR 4135090, DOI 10.3934/math.2019.2.316
- L. M. Milne-Thomson, Theoretical hydrodynamics, Macmillan and Co. Ltd, 1962.
- D. Riabouchinsky, On steady fluid motions with free surfaces, Proc. London Math. Sot. II. 19 (1921), 206–215.
- Ravindra Pethiyagoda, Timothy J. Moroney, and Scott W. McCue, Efficient computation of two-dimensional steady free-surface flows, Internat. J. Numer. Methods Fluids 86 (2018), no. 9, 607–624. MR 3767309, DOI 10.1002/fld.4469
- J. M. Vanden-Broeck and F. Dias, Free-surface flows with two stagnation points, J. Fluid Mech. 324 (1996), 393–406.
- J. M. Vanden-Broeck, The influence of capillarity on cavitating flow past a flat plate, Q. J. Mech. Appl. Math. 34 (1981), 465–473.
- Jean-Marc Vanden-Broeck, Gravity-capillary free-surface flows, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 2010. MR 2722683, DOI 10.1017/CBO9780511730276
- J.-M. Vanden-Broeck, Nonlinear capillary free-surface flows, J. Engrg. Math. 50 (2004), no. 4, 415–426. MR 2116762, DOI 10.1007/s10665-004-1769-2
- J. M. Vanden-Broeck, Cavitating flow of a fluid with surface tension past a circular cylinder, Phys. Fluids. A. 3 (1991), no. 2, 263–266.
- L. C. Wrobel, A simple and efficient BEM algorithm for planar cavity flows, Int. J. Numer. Meth. Fluids. 14 (1992), no. 5, 529–537.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2020):
76B07,
76B10,
65M38,
35Q35
Retrieve articles in all journals
with MSC (2020):
76B07,
76B10,
65M38,
35Q35
Additional Information
Abdelkader Laiadi
Affiliation:
Department of Mathematics and Computer Science, Biskra University, 7000, Biskra, Algeria
Address at time of publication:
Biskra University, Algeria
MR Author ID:
1071354
Email:
abdelkader.laiadi@univ-biskra.dz
Keywords:
Free surface flow,
cavitation flow,
Weber number,
surface tension,
Froude number
Received by editor(s):
November 5, 2021
Received by editor(s) in revised form:
January 20, 2022
Published electronically:
March 21, 2022
Article copyright:
© Copyright 2022
Brown University
