Scientific Research

An Academic Publisher

**Unsteady Spillway Flows by Singular Integral Operators Method** ()

The Singular Integral Operators Method (S.I.O.M.) is applied to the determination of the free-surface profile of an un-steady flow over a spillway, which defines a classical hydraulics problem in open channel flow. Thus, with a known flow rate Q, then the velocities and the elevations are computed on the free surface of the spillway flow. For the numerical evaluation of the singular integral equations both constant and linear elements are used. An application is finally given to the determination of the free-surface profile of a special spillway and comparing the numerical results with corresponding results by the Boundary Integral Equation Method (B.I.E.M.) and by using experiments.

Share and Cite:

E. Ladopoulos, "Unsteady Spillway Flows by Singular Integral Operators Method,"

*Engineering*, Vol. 4 No. 3, 2012, pp. 133-138. doi: 10.4236/eng.2012.43017.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | R. V. Southwell and G. Vaisey, “Relaxation Methods Applied to Engineering Problems: XIII, Fluid Motions Characterized by Free Streamlines,” Philosophical Trans- actions of the Royal Society of London. Series A, Ma- thematical and Physical Sciences, Vol. 240, No. 815, 1946, pp. 117-161. |

[2] | J. S. McNown, E.-Y. Hsu and C.-S. Yih, “Application of the Re-laxation Technique in Fluid Mechanics,” Trans- actions of ASCE, Vol. 120, 1955, pp. 650-686. |

[3] | J. J. Cassidy, “Irrota-tional Flow over Spillways of Finite Height,” Journal of the Engineering Mechanics Division, Vol. 91, No. 6, 1965, pp. 155-173 |

[4] | J. A. McCorquodale and C. Y. Li, “Finite Ele-ment Analysis of Sluice Gate Flow,” Engineering Journal, Vol. 54, No. , 1971, pp. 1-4. |

[5] | S. T. K. Chan, B. E. Larock and L. R. Hermann, “Free- Surface Ideal Fluid Flows by Finite Ele-ments,” Journal of the Hydraulics Division, Vol. 99, No. 3, 1973, pp. 959- 974. |

[6] | M. Ikegawa and K. Washizu, “Finite Element Method Applied to Analysis of Flow over a Spillway Crest,” In- ternational Journal for Numerical Methods in En-gineer- ing, Vol. 6, No. 2, 1973, pp. 179-189. doi:10.1002/nme.1620060204 |

[7] | L. T. Isaacs, “A Curved Triangular Finite Element for Potential Flow Problems,” Inter-national Journal for Nu- merical Methods in Engineering, Vol. 7, No. 3, 1973, pp. 337-344. doi:10.1002/nme.1620070310 |

[8] | L. T. Isaacs, “Numerical Solutionfor Flow under Sluice Gates,” Journal of the Hydrau-lics Division, Vol. 103, No. 5, 1977, pp. 473-481. |

[9] | B. E. Larock, “Flow over Gated Spillway Crests,” Proceedings of 14th Midwestern Mechanics Conference, Vol. 8, University of Oklahoma Press, Norman, 1975, pp. 437- 451. |

[10] | H. J. Diersch, A. Schirmer and K. F. Busch, “Analysis of Flows with Initially Unknown Discharge,” Journal of the Hydraulics Divi-sion, Vol. 103, No. 3, 1977, pp. 213-232. |

[11] | E. Varoglu and W. D. L. Finn, “Variable Domain Finite element Analysis of Free Surface Gravity Flow,” Com- putes & Fluids, Vol. 6, No. 2, 1978, pp. 103-114. doi:10.1016/0045-7930(78)90011-7 |

[12] | P. L. Betts, “A Variational Principle in Terms of Stream Function for Free-Surface Flows and Its Application to the Finite Element Method,” Computes & Fluids, Vol. 7, No. 2, 1979, pp. 145-153. doi:10.1016/0045-7930(79)90030-6 |

[13] | J. A. Liggett, “Loca-tion of Free Surface in Porous Me- dia,” Journal of the Hy-draulics Division, Vol. 103, No. 4, 1977, pp. 353-365. |

[14] | A. H.-D. Cheng, J. A. Liggeand P. L.-F. Liu, “Boundary Calcula-tions of Sluice and Spillway Flows,” Journal of the Hydraulics Division, Vol. 107, No. 4, 1981, pp. 1163- 1178. |

[15] | T. S. Strelkoff, “Solution of Highly Curvilinear Gravity Flows,” Journal of the Engineering Mechanics Division, 90, Vol. 90, No. 3, 1964, pp. 195-222. |

[16] | T. S. Strelkoff and M. S. Moayeri, “Patern of potential Flow in a Free Surface Overfall,” Journal of the Hydrau- lics Division, Vol. 96, No. 4, 1970, pp. 879-901. |

[17] | Y. Guo, X. Wen, C. Wu and D. Fang, “Numeri-cal Model- ling of Spillway Flow with Free Drop and Initially Un- known Discharge,” Journal of Hydraulic Research, Vol. 36, No. 5, 1998, pp. 785-801. doi:10.1080/00221689809498603 |

[18] | E. G. Ladopoulos, “Fi-nite—Part Singular Integro—Dif- ferential Equations Arising in Two-Dimensional Aerody- namics,” Archives of Mechanics, Vol. 41, No. , 1989, pp. 925-936. |

[19] | E. G. Ladopoulos, “Non-Linear Singular Integral Repre- sentation for Unsteady Inviscid Flowfields of 2-D Air- foils,” Mechanics Research Communications, Vol. 22, No. 1, 1995, pp. 25-34. doi:10.1016/0093-6413(94)00036-D |

[20] | E. G. Ladopoulos, “Non-Linear Singular Integral Compu- tational Analysis for Unsteady Flow Problems,” Renew- able Energy, Vol. 6, No. 8, 1995, pp. 901-906. doi:10.1016/0960-1481(95)00099-1 |

[21] | E. G. Ladopoulos, “Non-Linear Singular Integral Repre- sentation Analysis for Inviscid Flowfields of Unsteady Airfoils,” International Jour-nal of Non-linear Mechanics, Vol. 32, No. 4, 1997, pp. 377-384. doi:10.1023/A:1004246318082 |

[22] | E. G. Ladopoulos, “Non-Linear Multidimensional Singu- lar Integral Equations in 2-Dimensional Fluid Mechanics Analysis,” International Journal of Non-Linear Mechan- ics, Vol. 35, No. 4, 2000, pp. 701-708. |

[23] | E. G. Ladopoulos, “Non-Linear Unsteady Flow Problems by Multidimensional Singular Integral Representation Ana- lysis,” International Journal of Mathematics and Mathe- matical Sciences, Vol. 2003, No. 50, 2003, pp. 3203- 3216. |

[24] | E. G. Ladopoulos, “Non-Linear Two-Dimensional Aerodynamics by Multidimensional Singular Integral Compu- tational Analysis,” Forshung im Ingenieurwesen, Vol. 68, No. 2, 2003, pp. 105-110. doi:10.1007/s10010-003-0114-7 |

[25] | E. G. Ladopoulos, “Sin-gular Integral Equations, Linear and Non-Linear Theory and its Applications in Science and Engineering,” Springer, Berlin, 2000. |

[26] | E. G. Ladopoulos, “Singular Integral Equations in Poten- tial Flows of Open-Channel Transitions,” Computes & Fluids, Vol. 39, No. 9, 2010, pp. 1451-1455. doi:10.1016/j.compfluid.2010.04.013 |

[27] | E. G. Ladopoulos and V. A. Zisis, “Existence and Uni- queness for Non-Linear Singular Integral Equations Used in Fluid Mechanics,” Appli-cations of Mathematics, Vol. 42, No. 5, 1997, pp. 345-367. doi:10.1023/A:1023058024885 |

[28] | E. G. Ladopoulos and V. A. Zisis, “Non-Linear Finite-Part Singular Integral Equations Arising in Two-Dimensional Fluid Mechanics,” Nonlinear Analysis: Theory, Methods & Applications, 42, Vol. 42, No. 1, 2000, pp. 277-290. |

[29] | E. G. Ladopoulos, “On the Numerical Evaluation of the Singular Integral Equations Used in Two- and Three-Di- mensional Plasticity Problems,” Mechanics Research Com- munications, Vol. 14, No. 4, 1987, pp. 263-274. doi:10.1016/0093-6413(87)90039-5 |

[30] | E. G. Ladopoulos, “Singular Integral Representation of Three-Dimensional Plas-ticity Fracture Problem,” Theo- retical and Applied Fracture Mechanics, Vol. 8, No. 3, 1987, pp. 205-211. doi:10.1016/0167-8442(87)90047-4 |

[31] | E. G. Ladopoulos, “On the Numerical Solution of the Multidimensional Singular Integrals and Integral Equa- tions Used in the Theory of Linear Viscoelasticity,” In- ternational Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 3, 1988, pp. 561-574. doi:10.1155/S0161171288000675 |

[32] | E. G. Ladopoulos, “Singular Integral Operators Method for Two-Dimensional Plasticity Problems,” Computer & Structures, Vol. 33, No. 3, 1989, pp. 859-865. doi:10.1016/0045-7949(89)90260-5 |

[33] | E. G. Ladopoulos, “Cubature Formulas for Singular Integral Approximations Used in Three-Dimensional Elasticity,” Revue Roumaine des Sci-ences Techniques. Série de Mécanique Appliquée, Vol. 34, No. 4, 1989, pp. 377-389. |

[34] | E. G. Ladopoulos, “Singular Integral Operators Method for Three-Dimensional Elasto-Plastic Stress Analysis,” Com- puter & Structures, Vol. 38, No. 1, 1991, pp. 1-8. doi:10.1016/0045-7949(91)90117-5 |

[35] | E. G. Ladopoulos, “Singular Integral Operators Method for Two-Dimensional Elasto-Plastic Stress Analysis,” For- shung im Ingenieurwesen, Vol. 57, No. 5, 1991, pp. 152- 158. doi:10.1007/BF02561415 |

[36] | E. G. Ladopoulos, “Singular Integral Operators Method for Anisotropic Elastic Stress Analysis,” Computer & Structures, Vol. 48, No. 6, 1993, pp. 965-973. doi:10.1016/0045-7949(93)90431-C |

[37] | E. G. Ladopoulos, “3-D Elastostatics by Coupling Method of Singular Integral Equations with Finite Elements,” En- gineering Analysis with Boundary Elements, Vol. 26, No. 2, 2002, pp. 591-596 doi:10.1016/S0955-7997(02)00021-8 |

[38] | E. G. Ladopoulos, “Coupling of Singular Integral Equation Methods and Finite Elements in 2-D Elasticity,” For- shung im Ingenieurwesen, Vol. 69, No. 1, 2004, pp. 11-16. doi:10.1007/s10010-004-0131-1 |

[39] | V. T. Chow, “Open Channel Hydraulics,” McGraw-Hill, New York, 1959. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.