A comparison study on high-order bounded schemes: Flow of PTT-linear fluid in a lid-driven square cavity

Kerim Yapici

Korea-Australia Rheology Journal, Vol.24, No.1, 11-21, March, 2012

DOI10.1007/s13367-012-0002-5 Export Citation

A comparison study on high-order bounded schemes: Flow of PTT-linear fluid in a lid-driven square cavity

Kerim Yapici^†

Department of Chemical Engineering, Cumhuriyet University, 58140 Sivas, Turkey

E-mail:

In this computational study, the convergence, stability and order of accuracy of several different numerical schemes are assessed and compared. All of the schemes considered were developed using a normalized variable diagram. Two test cases are considered: (1) two-dimensional steady incompressible laminar flow of a Newtonian fluid in a square lid-driven cavity; and (2) creeping flow of a PTT-linear fluid in a lid-driven square cavity. The governing equations are discretized to varying degrees of refinement using uniform grids, and solved by using the finite volume technique. The momentum interpolation method (MIM) is employed to evaluate the face velocity. Coupled mass and momentum conservation equations are solved through an iterative SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm. Among the higher-order and bounded schemes considered in the present study, only the CLAM, COPLA, CUBISTA, NOTABLE, SMART and WACEB schemes provide a steady converged solution to the prescribed tolerance of 1×10.5 at all studied Weissenberg (We) numbers, using a very fine mesh structure. It is found that the CLAM, COPLA, CUBISTA, SMART and WACEB schemes provide about the same order of accuracy that is slightly higher than that of the NOTABLE scheme at low and high Weissenberg numbers. Moreover, flow structures formed in the cavity, i.e. primary vortex, are captured accurately up to We = 5 by all converged schemes.

Keywords:high-order schemes;lid-driven cavity;PTT-linear fluid;finite volume;collocated grid

[References]

Alves MA, Oliveira PJ, Pinho FT, Int. J. Numer. Methods Fluids, 4, 47 (2003)
Alves MA, Pinho FT, Oliveira PJ, J. Non-Newton. Fluid Mech., 93(2-3), 287 (2000)
Botella O, Peyret R, Comput. Fluids, 27(4), 421 (1998)
Chakravarthy SR, Osher S, AIAA J., 21, 1241 (1983)
Choi SK, Nam HY, Cho M, KSME J., 9, 240 (1995)
Coelho PJ, J. Quant. Spectrosc. Radiat. Transfer, 109, 189 (2008)
Cruz DOA, Pinho FT, Oliveira PJ, J. Non-Newton. Fluid Mech., 132(1-3), 28 (2005)
Darwish MS, Numer. Heat Transf. B-Fundam., 24, 353 (1993)
Erturk E, Gokcel C, Int. J. Numer. Methods Fluids, 50, 421 (2006)
Erturk E, Int. J. Numer. Methods Fluids, 60, 275 (2009)
Gaskell PH, Lau AKC, Int. J. Numer. Methods Fluids, 8, 617 (1988)
Harten A, J. Comput. Phys., 49, 357 (1983)
Hayase T, Humphrey JAC, Greif R, J. Comput. Phys., 98, 108 (1992)
Jasak H, Weller HG, Gosman AD, Int. J. Numer. Methods Fluids, 31, 431 (1999)
Khosla PK, Rubin SG, Comput. Fluids, 2, 207 (1974)
Leonard BP, Drummond JE, Int. J. Numer. Methods Fluids, 20, 421 (1995)
Leonard BP, Comp. Methods Appl. Mech. Eng., 19, 59 (1979)
Leonard BP, Int. J. Numer. Methods Fluids, 8, 1291 (1988)
Leonard BP, Comp. Methods Appl. Mech. Eng., 88, 17 (1991)
Marchi CH, Suero R, Araki LK, J. Braz. Soc. Mech. Sci. Eng., 31, 186 (2009)
Nacer B, David L, Pascal B, Gerard J, Heat Mass Transf., 43, 1075 (2007)
Ng KC, Yusoff MZ, Ng EYK, Numer. Heat Transf. B-Fundam., 50, 561 (2006)
Ng KC, Yusoff MZ, Ng EYK, Int. J. Numer. Methods Fluids, 53, 57 (2007)
Pascau A, Perez C, Sanchez D, Int. J. Numer. Methods Fluids, 5, 75 (1995)
Patankar SV, Spalding DB, Int. J. Heat Mass Transf., 15, 1787 (1972)
Patankar SV, 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York.
Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 2, 353 (1977)
Przulj V, Basara B, 2001, Bounded convection schemes for unstructured grids, AIAA Paper 2001-2593, Proceeding of the AIAA Computational Fluid Dynamic Conference, Anaheim, CA, U.S.A.
Roache PJ, Annu. Rev. Fluid Mech., 29, 123 (1997)
Schreiber R, Keller HB, J. Comput. Phys., 49, 310 (1983)
Song B, Liu GR, Lam KY, Amano RS, Int. J. Numer. Methods Fluids, 32, 881 (2000)
Sweby PK, SIAM J. Numer. Anal., 21, 995 (1984)
Van Leer B, J. Comput. Phys., 14, 361 (1974)
Van Leer B, J. Comput. Phys., 32, 101 (1979)
Versteeg HK, Malalasekera W, 1995, An introduction to computational fluid dynamics: The finite volume method, Prentice Hall.
Warming RF, Beam RM, AIAA J., 14, 1241 (1976)
Wei JJ, Yu WQ, Tao, Kawaguchi Y, Numer. Heat Transf. B-Fundam., 49, 585 (2006)
Wei JJ, Yu B, Tao WQ, Numer. Meth. Part Differ. Equ., 43, 19 (2003)
Yapici K, Karasozen B, Uludag Y, J. Non-Newton. Fluid Mech., 164(1-3), 51 (2009)
Zhu J, Rodi W, Comput. Methods Appl. Mech. Engng., 92, 225 (1991)
Zijlema M, Wesseling P, IJCFD, 9, 89 (1998)

[Referenced By]

[1] Alves MA, Oliveira PJ, Pinho FT, Int. J. Numer. Methods Fluids, 4, 47 (2003)

[2] Alves MA, Pinho FT, Oliveira PJ, J. Non-Newton. Fluid Mech., 93(2-3), 287 (2000)

[3] Botella O, Peyret R, Comput. Fluids, 27(4), 421 (1998)

[4] Chakravarthy SR, Osher S, AIAA J., 21, 1241 (1983)

[5] Choi SK, Nam HY, Cho M, KSME J., 9, 240 (1995)

[6] Coelho PJ, J. Quant. Spectrosc. Radiat. Transfer, 109, 189 (2008)

[7] Cruz DOA, Pinho FT, Oliveira PJ, J. Non-Newton. Fluid Mech., 132(1-3), 28 (2005)

[8] Darwish MS, Numer. Heat Transf. B-Fundam., 24, 353 (1993)

[9] Erturk E, Gokcel C, Int. J. Numer. Methods Fluids, 50, 421 (2006)

[10] Erturk E, Int. J. Numer. Methods Fluids, 60, 275 (2009)

[11] Gaskell PH, Lau AKC, Int. J. Numer. Methods Fluids, 8, 617 (1988)

[12] Harten A, J. Comput. Phys., 49, 357 (1983)

[13] Hayase T, Humphrey JAC, Greif R, J. Comput. Phys., 98, 108 (1992)

[14] Jasak H, Weller HG, Gosman AD, Int. J. Numer. Methods Fluids, 31, 431 (1999)

[15] Khosla PK, Rubin SG, Comput. Fluids, 2, 207 (1974)

[16] Leonard BP, Drummond JE, Int. J. Numer. Methods Fluids, 20, 421 (1995)

[17] Leonard BP, Comp. Methods Appl. Mech. Eng., 19, 59 (1979)

[18] Leonard BP, Int. J. Numer. Methods Fluids, 8, 1291 (1988)

[19] Leonard BP, Comp. Methods Appl. Mech. Eng., 88, 17 (1991)

[20] Marchi CH, Suero R, Araki LK, J. Braz. Soc. Mech. Sci. Eng., 31, 186 (2009)

[21] Nacer B, David L, Pascal B, Gerard J, Heat Mass Transf., 43, 1075 (2007)

[22] Ng KC, Yusoff MZ, Ng EYK, Numer. Heat Transf. B-Fundam., 50, 561 (2006)

[23] Ng KC, Yusoff MZ, Ng EYK, Int. J. Numer. Methods Fluids, 53, 57 (2007)

[24] Pascau A, Perez C, Sanchez D, Int. J. Numer. Methods Fluids, 5, 75 (1995)

[25] Patankar SV, Spalding DB, Int. J. Heat Mass Transf., 15, 1787 (1972)

[26] Patankar SV, 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York.

[27] Phan-Thien N, Tanner RI, J. Non-Newton. Fluid Mech., 2, 353 (1977)

[28] Przulj V, Basara B, 2001, Bounded convection schemes for unstructured grids, AIAA Paper 2001-2593, Proceeding of the AIAA Computational Fluid Dynamic Conference, Anaheim, CA, U.S.A.

[29] Roache PJ, Annu. Rev. Fluid Mech., 29, 123 (1997)

[30] Schreiber R, Keller HB, J. Comput. Phys., 49, 310 (1983)

[31] Song B, Liu GR, Lam KY, Amano RS, Int. J. Numer. Methods Fluids, 32, 881 (2000)

[32] Sweby PK, SIAM J. Numer. Anal., 21, 995 (1984)

[33] Van Leer B, J. Comput. Phys., 14, 361 (1974)

[34] Van Leer B, J. Comput. Phys., 32, 101 (1979)

[35] Versteeg HK, Malalasekera W, 1995, An introduction to computational fluid dynamics: The finite volume method, Prentice Hall.

[36] Warming RF, Beam RM, AIAA J., 14, 1241 (1976)

[37] Wei JJ, Yu WQ, Tao, Kawaguchi Y, Numer. Heat Transf. B-Fundam., 49, 585 (2006)

[38] Wei JJ, Yu B, Tao WQ, Numer. Meth. Part Differ. Equ., 43, 19 (2003)

[39] Yapici K, Karasozen B, Uludag Y, J. Non-Newton. Fluid Mech., 164(1-3), 51 (2009)

[40] Zhu J, Rodi W, Comput. Methods Appl. Mech. Engng., 92, 225 (1991)

[41] Zijlema M, Wesseling P, IJCFD, 9, 89 (1998)