Determination of the flow field in the phenomenon of falling of sediment particles in near-stagnant water using the two-dimensional Burger equation solution with the fully implicit finite difference method

Subject Areas : Subjects

Iman Rezaei ¹ (Civil Engineering Department, Persian Gulf University.)
M. واقفی ² (دانشیار گروه مهندسی عمران-سازه های هیدرولیکی، دانشکده مهندسی، دانشگاه خلیج فارس)
Hossein Rahideh ³ (Chemical Engineering Department, Persian Gulf University.)

Received: 2018-07-08 Accepted : 2020-05-12 Published : 2020-04-20

Keywords: Finite difference method, Finite element method, Partial differential equations, Burger equation, Velocity of falling of sediment particles,

Abstract :

Physical processes are dependent on various parameters that are modeled in their mathematical language with their own equation. Since some nonlinear partial differential equations are so difficult to solve, hence obtain their analytical solution except in certain conditions is not possible and such equations can be solved numerically. The equation that is considered in this research is the Burger equation in time-dependent nonlinear two-dimensional mode, which models the phenomenon of velocity of falling of particle in the stagnant or near-stagnant fluid, such as sedimentary water behind a dam. In this research, first, this equation discretized using the fully implicit finite difference method to solve the two-dimensional Burger equation, which is an unconditional stable method, then it programmed. Also, the accuracy of the equation solution results with another numerical method (finite element method) is compared which implies the consistency of the simpler method of finite difference to the more complex method of finite element. Numerical results have been obtained for different viscosities and times, and their role in the particle velocity has been examined parametrically. Generally, the results showed that increasing viscosity and time, lead to decreasing in fall of the particle velocity. Within the increasing the time, locus of the maximum velocities approach to the bottom of the bed and the end of the length in the depth and longitudinal directions, respectively. Also, the negative depth velocity (upward flow) was observed, especially at the edges near the bed, indicating that the particles are suspended in some positions and times.

References:

1)        Abdou M.A. and Soliman A.A. (2005) Variational iteration method for solving Burger's and coupled Burger's equations. J. comp. Appl. Math. 181, 245-251.

2)        Albeverio,S. Korshunova A. and Rozanova O. (1979) A probabilistic model associated with the pressureless gas dynamics. Bull. Sci. math. 137 351 -365.

3)        Arora G. and Singh B.K. (2013) Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method. Applied Mathematics and Computation 224, 166-177.

4)        Arminjon P. and Beauchamp, C. (1979) Numerical Solution of Burgers’ equations in two space dimensions. Computer Methods in Applied Mechanics and Engineering 19, 351-365.

5)        Bateman, H. (1915) Some recent researches on the motion of fluids. Monthly Weather Rev. 43, 163-170.

6)        Burger, J.M. (1948) A Mathematical Model Illustrating the Theory of Turbulence. Adv. in Appl. Mech. I, Academic Press, 171 -199.

7)        Burgers, J.M. (1939) Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Trans. R. Neth. Acad. Sci. 17, 1-53.

8)        Diaz J.E., Ramirez J. and Villa J. (2011) The numerical solution of a generalized Burgers–Huxley equation through a conditionally bounded and symmetry-preserving method. Computers & Mathematics with Applications 61, 3330-3342.

9)        Haq, S. Islam S. and Uddin M. (2009) A mesh-free method for the numerical solution of the KdV–Burgers equation. Applied Mathematical Modelling 33, 3442-3449.

10)     Khater, A.H. Temsah R.S. and Hassan M.M. (2008) A Chebyshev spectral collocation method for solving Burgers’-type equations. Journal of Computational and Applied Mathematics 222, 333-350.

11)     Kofman L. and Rage A.C. (1992) Modeling structures of knots in jet flows with the Burgers equation. Astrophys. J. 390, 359–364.

12)     Rady A.S., Osman E.S. and Khalfallah M. (2010) Multi-soliton solution, rational solution of the Boussinesq–Burgers equations. commun. nonlinear sci. numer. simulat. 15, 1172-1176.

13)     Rizun, V.I. and Engel’Brekht, Iu. K. (1975) Application of the Burgers’equation with a variable coefficient to the study of nonplanar wave transients. PMM Vol39, Ng3, 551-554.

14)     Shu C. and Richards B.E. (1992) Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stoks equation. Numerical Methods in Fluids 15, 791 -798.

15)     Shukla H.S., Tamsir M., Srivastava K. and Kumar J. (2016) Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method. AIP Advances 4, 117-134.

16)     Tasmir M. and Sirvastava V.K. (2011) A semi-implicit finite-difference approach for two-dimensional coupled Burgers’ equations. In. J. Sci. & Eng. Res. 2, 46-49.

17)     Vaghefi M., rahideh H., M.R. Golbahar Haghighi and A.K. khaksar (2012) Distributed Approximating Functional Approach to Burgers’ Equation using Element Differential Quadrature Method. J.Appl. Sci. Environ. Manage. 16, 143-149.

18)     Watanabe,S. Ishiwata,S. Kawamura K. and Oh H.G. (1997) Higher order solution of nonlinear waves. II. Shock wave described by Burgers equation. J. Phys. Soc. Jpn. 66 984–987.

19)     Wazwaz A.M. (2014) A study on a (2+ 1)-diensional and a (3+ 1)-dimensional generalized Burgers equation. Applied Mathematics Letters 31, 41-45.

20)     Wei G.W., Zhang D.S., Kouri D.J. and Hoffman D.K. (1998) Distributed approximating Functional approach to Burgers’ equation in one and two space dimentions. Computer Physics Communications 111, 93 - 109.

21)     Zaki, S.I. (2000) A quintic B-spline finite elements scheme for the KdVB equation.   Comput. Methods Appl. Mech. Engrg. 188, 121-134.

22)     Zhanlav, T. Chuluunbaatar O. and Ulziibayar V. (2015) Higher-order accurate numerical solution of unsteady Burgers’ equation. Applied Mathematics and Computation 250, 701-707.

23)     Zheng,Q. Zhao X. and Liu Y. (2017) A novel finite difference scheme for Burgers' equation on unbounded domains. Applied Numerical Mathematics 111, 1-16.