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

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Abstract

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.

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References

  • Alves, M. A., P. J. Oliveira, and F. T. Pinho, 2003, A convergent and universally bounded interpolation scheme for the treatment of advection, Int. J. Numer. Meth. Fluids 4, 47–75.

    Article  Google Scholar 

  • Alves, M. A., F. T. Pinho, and P. J. Oliveira, 2000, Effect of a high-resolution differencing scheme on finite-volume predictions of viscoelastic flows, J. Non-Newtonian Fluid Mech. 93, 287–314.

    Article  CAS  Google Scholar 

  • Botella, O. and R. Peyret, 1998, Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids 27(4), 421–433.

    Article  Google Scholar 

  • Chakravarthy, S. R. and S. Osher, 1983, High-resolution of the OSHER upwind scheme for the Euler equations, AIAA J. 21, 1241–1248.

    Article  Google Scholar 

  • Choi, S. K., H. Y. Nam, and M. Cho, 1995, A high resolution and bounded convection scheme, KSME J. 9, 240–250.

    Google Scholar 

  • Coelho, P. J., 2008, A comparison of spatial discretization schemes for differential solution methods of the radiative transfer equation, J. Quant. Spectrosc. Radiat. Transfer 109, 189–200.

    Article  CAS  Google Scholar 

  • Cruz, D. O. A., F. T. Pinho, and P. J. Oliveira, 2005, Analytical solution for fully developed laminar flow of some viscoelastic liquids with a Newtonian solvent contribution, J. Non-Newtonian Fluid Mech. 132, 28–35.

    Article  CAS  Google Scholar 

  • Darwish, M. S., 1993, A new high-resolution scheme based on the normalized variable formulation, Numer. Heat Transfer, Part B 24, 353–373.

    Article  CAS  Google Scholar 

  • Erturk, E. and C. Gökçel, 2006, Fourth order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids 50, 421–436.

    Article  Google Scholar 

  • Erturk, E., 2009, Discussion on driven cavity flow, Int. J. Numer. Meth. Fluids 60, 275–294.

    Article  Google Scholar 

  • Gaskell, P. H. and A. K. C. Lau, 1988, Curvature-compensated convective tansport: SMART, a new boundedness preserving transport algorithm, Int J Numer Meth Fluids 8, 617–641.

    Article  Google Scholar 

  • Harten, A., 1983, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49, 357–393.

    Article  Google Scholar 

  • Hayase, T., J. A. C. Humphrey, and R. Greif, 1992, A consistently formulated quick scheme for fast and stable convergence using finite-volume iterative calculation procedures, J. Comp. Phys. 98, 108–118.

    Article  Google Scholar 

  • Jasak, H., H. G. Weller, and A. D. Gosman, 1999, High resolution NVD differencing scheme for arbitrarily unstructured meshes, Int. J. Numer. Meth. Fluids, 31, 431–449.

    Article  Google Scholar 

  • Khosla, P. K. and S. G. Rubin, 1974, A diagonally dominant second order accurate implicit scheme, Comput. Fluids 2, 207–209.

    Article  Google Scholar 

  • Leonard, B. P. and J. E. Drummond, 1995, Why you should not use ‘Hybrid’, ‘Power-Law’ or related exponential schemes for convective modeling: there are much better alternatives, Int. J. Numer. Meth. Fluids 20, 421–442.

    Article  Google Scholar 

  • Leonard, B. P., 1979, A stable and accurate convective modeling procedure based on quadratic interpolation, Comp. Methods Appl. Mech. Eng. 19, 59–98.

    Article  Google Scholar 

  • Leonard, B. P., 1988, Simple high-accuracy resolution program for convective modeling of discontinuities, Int. J. Numer. Meth. Fluids 8, 1291–1318.

    Article  Google Scholar 

  • Leonard, B. P., 1991, The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection, Comp. Methods Appl. Mech. Eng. 88, 17–74.

    Article  Google Scholar 

  • Marchi, C. H., R. Suero, and L. K. Araki, 2009, The lid-driven square cavity flow: Numerical solution with a 1024×1024 grid, J. Braz. Soc. Mech. Sci. Eng. 31, 186–198.

    Article  Google Scholar 

  • Nacer, B., L. David, B. Pascal, and J. Gérard, 2007, Contribution to the improvement of the QUICK scheme for the resolution of the convection-diffusion problems, Heat Mass Transfer 43, 1075–1085.

    Article  Google Scholar 

  • Ng, K. C., M. Z. Yusoff, and E. Y. K. Ng, 2006, Parametric study of an improved GAMMA differencing scheme based on normalized variable formulation for low-speed flow with artificial compressibility technique, Numer. Heat Transfer, Part B 50, 561–584.

    Article  Google Scholar 

  • Ng, K. C., M. Z. Yusoff, and E. Y. K. Ng, 2007, Higher-order bounded differencing schemes for compressible and incompressible flows, Int. J. Numer. Meth. Fluids 53, 57–80.

    Article  Google Scholar 

  • Pascau, A., C. Perez, and D. Sánchez, 1995, A well-behaved scheme to model strong convection in general transport equation, Int. J. Numer. Methods for Heat Fluid Flow 5, 75–87.

    Article  CAS  Google Scholar 

  • Patankar, S. V. and D. B. Spalding, 1972, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transfer 15, 1787–1806.

    Article  Google Scholar 

  • Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York.

    Google Scholar 

  • Phan-Thien, N. and R. I. Tanner, 1977, A new constitutive equation derived from network theory, J. Non-Newtonian Fluid Mech. 2, 353–365.

    Article  Google Scholar 

  • Przulj, V. and B. Basara, 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, P. J., 1997, Quantification of uncertainty in computational fluid dynamics, Annu. Rev. Fluid Mech. 29,123–160.

    Article  Google Scholar 

  • Schreiber, R. and H. B. Keller, 1983, Driven cavity flows by efficient numerical techniques, J. Compt. Phys. 49, 310–333.

    Article  Google Scholar 

  • Song, B., G. R. Liu, K. Y. Lam, and R. S. Amano, 2000, On a higher-order bounded discretization scheme, Int. J. Numer. Meth. Fluids 32, 881–897.

    Article  CAS  Google Scholar 

  • Sweby, P. K., 1984, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21, 995–1011.

    Article  Google Scholar 

  • Van Leer, B., 1974, Towards the ultimate conservation difference scheme. II. Monotonicity and conservation combined in second-order scheme, J. Comput. Phys. 14, 361–370.

    Article  Google Scholar 

  • Van Leer, B., 1979, Towards the ultimate conservation difference scheme. V. A second-order sequel to Godunov’s Method, J. Comput. Phys. 32, 101–136.

    Article  Google Scholar 

  • Versteeg, H. K. and W. Malalasekera, 1995, An introduction to computational fluid dynamics: The finite volume method, Prentice Hall.

  • Warming, R. F. and R. M. Beam 1976, Upwind second-order difference schemes and applications in aerodynamic flows, AIAA J. 14, 1241–1249.

    Article  Google Scholar 

  • Wei, J. J., Yu, W. Q. Tao, and Y. Kawaguchi, 2006, A new general convective boundness criterion, Numer. Heat Transfer, Part B 49, 585–598.

    Article  Google Scholar 

  • Wei, J. J., B. Yu, and W. Q., Tao, 2003, A new high-order-accurate and bounded scheme for incompressible flow, Numer. Heat Transfer, Part B 43, 19–41.

    Article  Google Scholar 

  • Yapici, K., B. Karasozen, and Y. Uludag, 2009, Finite volume simulation of viscoelastic laminar flow in a lid-driven cavity, J. Non-Newtonian Fluid Mech. 164, 51–65.

    Article  CAS  Google Scholar 

  • Zhu, J. and W. Rodi, 1991, A low dispersion and bounded convection scheme, Comput. Methods Appl. Mech. Engng. 92, 225–232.

    Article  Google Scholar 

  • Zijlema, M. and P. Wesseling, 1998, Higher-order flux-limiting schemes for the finite volume computation of incompressible flow, IJCFD 9, 89–109.

    Google Scholar 

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Yapici, K. A comparison study on high-order bounded schemes: Flow of PTT-linear fluid in a lid-driven square cavity. Korea-Aust. Rheol. J. 24, 11–21 (2012). https://doi.org/10.1007/s13367-012-0002-5

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  • DOI: https://doi.org/10.1007/s13367-012-0002-5

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