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Research Papers: Techniques and Procedures

Application of an Iterative High Order Difference Scheme Along With an Explicit System Solver for Solution of Stream Function-Vorticity Form of Navier–Stokes Equations

[+] Author and Article Information
Parviz Ghadimi

e-mail: pghadimi@aut.ac.ir

Mehdi Yousefi Fard

e-mail: yousefifard@aut.ac.ir

Abbas Dashtimanesh

e-mail: a.dashtimanesh@aut.ac.ir
Department of Marine Technology,
Amirkabir University of Technology,
Hafez Avenue, No. 424,
P.O. Box 15875-4413
Tehran, Iran

Manuscript received July 26, 2012; final manuscript received December 23, 2012; published online February 22, 2013. Assoc. Editor: Chunill Hah.

J. Fluids Eng 135(4), 041401 (Feb 22, 2013) (11 pages) Paper No: FE-12-1347; doi: 10.1115/1.4023295 History: Received July 26, 2012; Revised December 23, 2012

This paper describes the general convection-diffusion equation in 2D domain based on a particular fourth order finite difference method. The current fourth-order compact formulation is implemented for the first time, which offers a semi-explicit method of solution for the resulting equations. A nine point finite difference scheme with uniform grid spacing is also put into action for discretization purpose. The proposed numerical model is based on the Navier–Stokes equations in a stream function-vorticity formulation. The fast convergence characteristic can be mentioned as an advantage of this scheme. It combines the enhanced Fournié's fourth order scheme and the expanded fourth order boundary conditions, while offering a semi-explicit formulation. To accomplish this, some coefficients which do not influence the solutions are also omitted from Fournié's formulation. Consequently, very accurate results can be acquired with a relatively coarse mesh in a short time. The robustness and accuracy of the proposed scheme is proved using the benchmark problems of flow in a driven square cavity at medium and relatively high Reynolds numbers, flow over a backward-facing step, and flow in an L-shaped cavity.

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Figures

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Fig. 1

Labeling of the nine grid points

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Fig. 2

Geometry and boundary conditions of the square cavity

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Fig. 3

Vorticity and stream function contours: Re = 1000

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Fig. 4

(a) Profile of u along the cavity vertical centerline, and (b) profile of v along the horizontal centerline for Re = 1000

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Fig. 5

Vorticity and stream function contours: Re = 5000

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Fig. 6

(a) Profile of u along the cavity vertical centerline, and (b) profile of v along the horizontal centerline for Re = 5000, N = 256

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Fig. 7

(a) Profile of u along the cavity vertical centerline, and (b) profile of v along the horizontal centerline for Re = 5000, N = 512

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Fig. 8

Vorticity and stream function contours: Re = 10,000

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Fig. 9

(a) Profile of u along the cavity vertical centerline, and (b) profile of v along the horizontal centerline for Re = 10,000, N = 512

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Fig. 19

Reattachment length

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Fig. 18

Stream function contours: Re = 100–1000

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Fig. 17

Geometry and boundary conditions of backward-facing step flow problem

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Fig. 16

(a) Profile of u along CL1, and (b) profile of v along CL2 for Re = 1000

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Fig. 15

Vorticity and stream function contours: Re = 1000

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Fig. 14

(a) Profile of u along CL1, and (b) profile of v along CL2 for Re = 100

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Fig. 13

Vorticity and stream function contours: Re = 100

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Fig. 12

Geometry of the L-shaped cavity

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Fig. 11

(a) Profile of u along the cavity vertical centerline, and (b) profile of v along the horizontal centerline for Re = 20,000, N = 1024

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Fig. 10

Vorticity and stream function contours: Re = 20,000

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