Numerical Solutions of Two-Dimensional Vorticity Transport Equation Using Crank-Nicolson Method

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Maan A. Rasheed
Suad Naji Kadhim

Abstract

This paper is concerned with the numerical solutions of the vorticity transport equation (VTE) in two-dimensional space with homogenous Dirichlet boundary conditions. Namely, for this problem, the Crank-Nicolson finite difference equation is derived.  In addition, the consistency and stability of the Crank-Nicolson method are studied. Moreover, a numerical experiment is considered to study the convergence of the Crank-Nicolson scheme and to visualize the discrete graphs for the vorticity and stream functions. The analytical result shows that the proposed scheme is consistent, whereas the numerical results show that the solutions are stable with small space-steps and at any time levels.

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Rasheed MA, Kadhim SN. Numerical Solutions of Two-Dimensional Vorticity Transport Equation Using Crank-Nicolson Method. Baghdad Sci.J [Internet]. [cited 2021Dec.4];19(2):0321. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/5807
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References

Alabedalhadi M, Al-Smadi M, Al-Omari S, Baleanu D, Momani S. Structure of optical soliton solution for nonlinear resonant space-time Schrödinger equation in conformablesense with full nonlinearity term. Phys. Scr. 2020; 95 (10): 105215.

Al-Smadi M, Abu Arqub O. Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Appl. Math. Comput. 2019; 342(C): 280-294.

Al-Smadi M, Freihat A, Khalil H, Momani S, Khan R A. Numerical multistep approach for solving fractional partial differential equations. Int. J. Comput. Methods. 2017; 14 (3): 1750029.

Al-Smadi M, Abu Arqub O, Momani S. Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense. Phys. Scr. 2020; 95 (7):075218.

Al-Smadi M, Abu Arqub O, Hadid S. An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative. Commun. Theor. Phys. 2020; 72 (8): 085001.

Tu J, Yeoh GH, Liu C.Computational fluid dynamics: A Practical Approach. 3d edition, Butterworth–Heinemann, UK; 2018.

Kaushik A. Numerical study of 2D incompressible flow in a rectangular domain using chorin’s projection method at high Reynolds number. Int. j. math. eng. manag. sci. 2019; 4(1): 157–169.

Pozrikidis C. Equation of motion and vorticity transport. In: Fluid Dynamics. Springer, Boston, MA; 2017.

Speziale C G. On the advantages of the vorticity-velocity formulation of the equations of fluid dynamics. J. Comput. Phys. 1987; 73: 476-480.

Tezduyar T E, Liou J, Ganjoo D K, Behr M. Solution techniques for the vorticity-streamfunction formulation of two-dimensional unsteady

incompressible flows. Int. J. Numer. Methods Fluids, 1990; 11(5):515–539

Dennis SC. The numerical solution of the vorticity transport equation. InProceedings of the third international conference on numerical methods in fluid mechanics 1973 (pp. 120-129). Springer, Berlin, Heidelberg.

Joseph M. Finite difference representations of vorticity transport. Comput Methods Appl Mech Eng. 1983 Aug 1;39(2):107-16.

Napolitano M , Pascazio G. A numerical method for the vorticity-velocity Navier-Stokes equations in two and three dimensions. Computers & Fluids. 1991 Jan 1;19(3-4):489-95

Lo D C. Murugesan K , Young D L. Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method. Int. J. Numer. Methods Fluids. 2005; 47(12): 1469-1487.

Ambethkar V , Kumar M , Srivastava M K. Numerical solutions of 2-d unsteady incompressible flow in a driven square cavity using streamfunction-vorticity formulation. Int. J. Appl. Math. 2016; 29 (6): 727-757.

Rasheed M A, Balasim A T, Jameel A F. Some results for the vorticity transport equation by using A.D.I scheme, AIP Conference Proceedings. 2019; 2138, 030031; doi.org/10.1063/1.5121068

Ravnik J, Tibaut J. Boundary-domain Integral method For vorticity transport equation with variable viscosity. Int. J. Comp. Meth. and Exp. Meas. 2018; 6(6): 1087–1096.

Mitchell A R. Computational methods in partial differential equations, Wiley, London; 1969.