The formulation and implementation of a high-order Cartesian cut-stencil finite difference method (CCST-FDM) to 2-D steady incompressible fluid flow in regular and irregular domains is considered in this paper. The CCST-FDM is capable of simulating flows in complex geometries by employing 1-D quadratic transformation functions to map any (uniform or non-uniform) physical stencil to a uniform computational stencil. In this work, the CCST-FDM is combined with compact high-order (HO) Padé-Hermitian approximations to produce HO CCST-FD schemes. Two different high-order (globally 4th-order) accurate schemes are formulated for the CCST-FDM. Using the streamfunction-vorticity formulation of the Navier-Stokes equations, low and high-order solutions are computed for lid-driven flows in four different geometries, namely a square cavity and three irregular regions, namely L-shape, skewed quadrilateral and triangular cavities. Results from these geometries are compared to earlier studies for various Reynolds numbers. It is shown that the high-order CCST-FDMs can achieved higher accuracy on coarser grid than other high-order methods.
References
[1]
Hoffmann, K.A. and Chiang, S.T. (2004) Computational Fluid Dynamics. Vol. 1. 4th Edition, Engineering Education System, Wichita, Kansas, USA.
[2]
Ghia, U., Ghia, K.N. and Shin, C.T. (1982) High Resolutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method. Journal of Computational Physics, 48, 387-411. https://doi.org/10.1016/0021-9991(82)90058-4
[3]
Spotz, W.F. (1998) Accuracy and Performance of Numerical Wall Boundary Conditions for Steady, 2D, Incompressible Streamfunction Vorticity. International Journal for Numerical Methods in Fluids, 28, 737-757. https://doi.org/10.1002/(sici)1097-0363(19980930)28:4<737::aid-fld744>3.3.co;2-c
[4]
Yu, P.X., Tian, Z.F. and Zhang, H. (2017) A Rational High-Order Compact Difference Method for the Steady-State Stream Function-Vorticity Formulation of the Navier–Stokes Equations. Computers & Mathematics with Applications, 73, 1461-1484. https://doi.org/10.1016/j.camwa.2017.01.024
[5]
Yu, Q., Xu, H., Liao, S. and Yang, Z. (2019) A Novel Homotopy-Wavelet Approach for Solving Stream Function-Vorticity Formulation of Navier-Stokes Equations. Communications in Nonlinear Science and Numerical Simulation, 67, 124-151. https://doi.org/10.1016/j.cnsns.2018.07.001
[6]
Ferziger, J.H. (1981) Numerical Methods for Engineering Application. Wiley.
[7]
Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences, Hemisphere Publishing.
[8]
Versteeg, H.K. and Malalasekera, W. (2007) An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education Ltd.
[9]
Reddy, J.N. (1984) An Introduction to the Finite Element Method. McGraw-Hill.
[10]
Esmaeilzadeh, M., Barron, R.M. and Balachandar, R. (2020) Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part I: Concepts and Fundamentals. NumericalMathematics: Theory, MethodsandApplications, 13, 881-907. https://doi.org/10.4208/nmtma.oa-2019-0143
[11]
Esmaeilzadeh, M. (2016) A Cartesian Cut-Stencil Method for the Finite Difference Solution of PDEs in Complex Domains. Ph.D. Dissertation, University of Windsor.
[12]
Choo, S.M. and Chung, S.K. (2000) High-Order Perturbation-Difference Scheme for a Convection-Diffusion Problem. Computer Methods in Applied Mechanics and Engineering, 190, 721-732. https://doi.org/10.1016/s0045-7825(99)00444-2
[13]
Liu, D., Kuang, W. and Tangborn, A. (2009) High-Order Compact Implicit Difference Methods for Parabolic Equations in Geodynamo Simulation. Advances in Mathematical Physics, 2009, Article 568296. https://doi.org/10.1155/2009/568296
[14]
Zhang, W. and Jiang, J. (2017) A New Family of Fourth-Order Locally One-Dimensional Schemes for the Three-Dimensional Wave Equation. Journal of Computational and Applied Mathematics, 311, 130-147. https://doi.org/10.1016/j.cam.2016.07.020
[15]
Adam, Y. (1975) A Hermitian Finite Difference Method for the Solution of Parabolic Equations. Computers & Mathematics with Applications, 1, 393-406. https://doi.org/10.1016/0898-1221(75)90041-3
[16]
Chu, P.C. and Fan, C. (2000) A Three-Point Sixth-Order Staggered Combined Compact Difference Scheme. Mathematical and Computer Modelling, 32, 323-340. https://doi.org/10.1016/s0895-7177(00)00138-2
[17]
Kong, L., Zhu, P., Wang, Y. and Zeng, Z. (2019) Efficient and Accurate Numerical Methods for the Multidimensional Convection-Diffusion Equations. Mathematics and Computers in Simulation, 162, 179-194. https://doi.org/10.1016/j.matcom.2019.01.014
[18]
Patel, K.S. and Mehra, M. (2020) Fourth Order Compact Scheme for Space Fractional Advection-Diffusion Reaction Equations with Variable Coefficients. Journal of Computational and Applied Mathematics, 380, Article 112963. https://doi.org/10.1016/j.cam.2020.112963
[19]
Esmaeilzadeh, M. and Barron, R.M. (2022) Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part II: Higher-Order Schemes. Numerical Mathematics: Theory, Methods and Applications, 15, 819-850. https://doi.org/10.4208/nmtma.oa-2021-0129
[20]
Zhong, X. (1998) High-Order Finite-Difference Schemes for Numerical Simulation of Hypersonic Boundary-Layer Transition. Journal of Computational Physics, 144, 662-709. https://doi.org/10.1006/jcph.1998.6010
[21]
Zhong, X. and Tatineni, M. (2003) High-Order Non-Uniform Grid Schemes for Numerical Simulation of Hypersonic Boundary-Layer Stability and Transition. Journal of Computational Physics, 190, 419-458. https://doi.org/10.1016/s0021-9991(03)00282-1
[22]
Erturk, E. (2009) Comparison of Wide and Compact Fourth-Order Formulations of the Navier-Stokes Equations. International Journal for Numerical Methods in Fluids, 60, 992-1010. https://doi.org/10.1002/fld.1920
[23]
Li, M., Tang, T. and Fornberg, B. (1995) A Compact Fourth-Order Finite Difference Scheme for the Steady Incompressible Navier-Stokes Equations. International Journal for Numerical Methods in Fluids, 20, 1137-1151. https://doi.org/10.1002/fld.1650201003
[24]
Gupta, M.M. and Kalita, J.C. (2005) A New Paradigm for Solving Navier-Stokes Equations: Streamfunction-Velocity Formulation. Journal of Computational Physics, 207, 52-68. https://doi.org/10.1016/j.jcp.2005.01.002
[25]
Pandit, S.K., Kalita, J.C. and Dalal, D.C. (2008) A Fourth-Order Accurate Compact Scheme for the Solution of Steady Navier-Stokes Equations on Non-Uniform Grids. Computers&Fluids, 37, 121-134. https://doi.org/10.1016/j.compfluid.2007.04.002
[26]
Thom, A. (1928) An Investigation of Fluid Flow in Two Dimensions. Reports and Memoranda, No. 1194. Aeronautical Research Committee.
[27]
Thom, A. (1933) The Flow Past Circular Cylinders at Low Speeds. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 141, 651-669. https://doi.org/10.1098/rspa.1933.0146
[28]
Gresho, P.M. (1991) Incompressible Fluid Dynamics: Some Fundamental Formulation Issues. Annual Review of Fluid Mechanics, 23, 413-453. https://doi.org/10.1146/annurev.fl.23.010191.002213
[29]
Sahin, M. and Owens, R.G. (2003) A Novel Fully Implicit Finite Volume Method Applied to the Lid-Driven Cavity Problem—Part I: High Reynolds Number Flow Calculations. International Journal for Numerical Methods in Fluids, 42, 57-77. https://doi.org/10.1002/fld.442
[30]
Benjamin, A.S. and Denny, V.E. (1979) On the Convergence of Numerical Solutions for 2-D Flows in a Cavity at Large Re. Journal of Computational Physics, 33, 340-358. https://doi.org/10.1016/0021-9991(79)90160-8
[31]
Briley, W.R. (1971) A Numerical Study of Laminar Separation Bubbles Using the Navier-Stokes Equations. Journal of Fluid Mechanics, 47, 713-736. https://doi.org/10.1017/s0022112071001332
[32]
Gupta, M.M. (1991) High Accuracy Solutions of Incompressible Navier-Stokes Equations. Journal of Computational Physics, 93, 343-359. https://doi.org/10.1016/0021-9991(91)90188-q
[33]
Orszag, S.A. and Israeli, M. (1974) Numerical Simulation of Viscous Incompressible Flows. Annual Review of Fluid Mechanics, 6, 281-318. https://doi.org/10.1146/annurev.fl.06.010174.001433
[34]
Yu, W. Q. Tao, D. S. Zhang, Q. W. W, B. (2001) Discussion on Numerical Stability and Boundedness of Convective Discretized Scheme. Numerical Heat Transfer, Part B: Fundamentals, 40, 343-365. https://doi.org/10.1080/104077901317091721
[35]
Paramane, S.B. and Sharma, A. (2008) Consistent Implementation and Comparison of FOU, CD, SOU, and QUICK Convection Schemes on Square, Skew, Trapezoidal, and Triangular Lid-Driven Cavity Flow. Numerical Heat Transfer, Part B: Fundamentals, 54, 84-102. https://doi.org/10.1080/10407790802122519
[36]
Pandit, S.K. (2008) On the Use of Compact Streamfunction-Velocity Formulation of Steady Navier-Stokes Equations on Geometries Beyond Rectangular. Journal of Scientific Computing, 36, 219-242. https://doi.org/10.1007/s10915-008-9186-8
[37]
Nishida, H. and Satofuka, N. (1992) Higher-Order Solutions of Square Driven Cavity Flow Using a Variable-Order Multi-Grid Method. International Journal for Numerical Methods in Engineering, 34, 637-653. https://doi.org/10.1002/nme.1620340215
[38]
Erturk, E. (2009) Discussions on Driven Cavity Flow. International Journal for Numerical Methods in Fluids, 60, 275-294. https://doi.org/10.1002/fld.1887
[39]
Oosterlee, C.W., Wesseling, P., Segal, A. and Brakkee, E. (1993) Benchmark Solutions for the Incompressible Navier-Stokes Equations in General Co-Ordinates on Staggered Grids. International Journal for Numerical Methods in Fluids, 17, 301-321. https://doi.org/10.1002/fld.1650170404
[40]
Perng, C.Y. and Street, R.L. (1991) A Coupled Multigrid‐domain‐splitting Technique for Simulating Incompressible Flows in Geometrically Complex Domains. InternationalJournalforNumericalMethodsinFluids, 13, 269-286. https://doi.org/10.1002/fld.1650130302
[41]
Erturk, E. and Dursun, B. (2007) Numerical Solutions of 2-D Steady Incompressible Flow in a Driven Skewed Cavity. ZAMM—Journal of Applied Mathematics and Mechanics, 87, 377-392. https://doi.org/10.1002/zamm.200610322
[42]
Erturk, E. and Gokcol, O. (2007) Fine Grid Numerical Solutions of Triangular Cavity Flow. The European Physical Journal Applied Physics, 38, 97-105. https://doi.org/10.1051/epjap:2007057
[43]
Shklyar, A. and Arbel, A. (2003) Numerical Method for Calculation of the Incompressible Flow in General Curvilinear Co-Ordinates with Double Staggered Grid. International Journal for Numerical Methods in Fluids, 41, 1273-1294. https://doi.org/10.1002/fld.427
[44]
Louaked, M., Hanich, L. and Nguyen, K.D. (1997) An Efficient Finite Difference Technique for Computing Incompressible Viscous Flows. International Journal for Numerical Methods in Fluids, 25, 1057-1082. https://doi.org/10.1002/(sici)1097-0363(19971115)25:9<1057::aid-fld605>3.0.co;2-j
[45]
McQuain, W.D., Ribbens, C.J., Wang, C.Y. and Watson, L.T. (1994) Steady Viscous Flow in a Trapezoidal Cavity. Computers & Fluids, 23, 613-626. https://doi.org/10.1016/0045-7930(94)90055-8
[46]
Ahmed, M. and Kuhlmann, H.C. (2012) Flow Instability in Triangular Lid-Driven Cavities with Wall Motion Away from a Rectangular Corner. Fluid Dynamics Research, 44, Article 025501. https://doi.org/10.1088/0169-5983/44/2/025501
[47]
Munir, F.A., Che Sidik, N.A., Mohd, M.I., et al. (2011) Application of Lattice Boltzmann Method in Predicting Flow of Shear Driven Cavities. Journal of Mechanical Engineering and Technology, 3, 55-70.
