This article presents a two-dimensional steady viscous flow simulation past circular and square cylinders at low Reynolds numbers (based on the diameter) by the finite volume method with a non-orthogonal body-fitted grid. Diffusive fluxes are discretized using central differencing scheme, and for convective fluxes upwind and central differencing schemes are blended using a ‘deferred correction’ approach. A simplified pressure correction equation is derived, and proper under-relaxation factors are used so that computational cost is reduced without adversely affecting the convergence rate. The governing equations are expressed in Cartesian velocity components and solution is carried out using the SIMPLE algorithm for collocated arrangement of variables. The mesh yielding grid-independent solution is then utilized to study, for the very first time, the effect of the Reynolds number on the separation bubble length, separation angle, and drag coefficients for both circular and square cylinders. Finally, functional relationships between the computed quantities and Reynolds number (Re) are proposed up to Re = 40. It is found that circular cylinder separation commences between Re= 6.5-6.6, and the bubble length, separation angle, total drag vary as Re, Re^{−0.5}, Re^{−0.5} respectively. Extrapolated results obtained from the empirical relations for the circular cylinder show an excellent agreement with established data from the literature. For a square cylinder, the bubble length and total drag are found to vary as Re and Re^{−0.666}, and are greater than these for a circular cylinder at a given Reynolds number. The numerical results substantiate that a square shaped cylinder is more bluff than a circular one.
[1] Strouhal V. (1878): Ueber eine besondere Art der Tonerregung. − Annalen der Physik und Chemie, vol.241, pp.216-251.
[2] Nisi H. and Porter A.W. (1923): Philos. Mag., vol.46, pp.754.
[3] Taneda S. (1956): Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. − J. Phys. Soc. Japan, vol.11, pp.302–307.
[4] Tritton D.J. (1959): Experiments on the flow past a circular cylinder at low Reynolds numbers. − J. Fluid Mech. vol.6, pp.547-567.
[5] Grove A.S., Shair F.H., Peterson E.E. and Acrivos A. (1964): An experimental investigation of the steady separated flow past a circular cylinder. − J. Fluid Mech. vol.19, pp.60-80.
[6] Acrivos A., Snowden D.D., Grove A.S. and Peterson E.E. (1965): The steady separated flow pasta circular cylinder at large Reynolds numbers. − J. Fluid Mech., vol.21, pp.737-760.
[7] Acrivos A., Leal L.G., Snowden D.D. and Pan F. (1968): Further experiments on steady separated flows past bluff objects. − J. Fluid Mech., vol.34, pp.25-48.
[8] Nishioka M. and Sato H. (1974): Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers. − J. Fluid Mech., vol.65, pp.97-112.
[9] Coutanceau M. and Bouard R. (1977): Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow. − J. Fluid Mech., vol.79, pp.231-256.
[10] Thom A. (1933): The flow past circular cylinders at low speeds. − Proc. R. Soc. Lond. A, vol.141, pp.651-669.
[11] Kawaguti M. (1953): Numerical solution of the Navier-Stokes equations for the flow around a circular cylinder at Reynolds number 40. − J. Phys. Soc. Jpn, vol.8, pp.747-757.
[12] Apelt C.J. (1961): The steady flow of a viscous fluid past a circular cylinder at Reynolds numbers40 and 44. − Aeronaut. Res. Counc. Lond. R & M, vol.3175, pp.1-28.
[13] Kawaguti M. and Jain P. (1966): Numerical study of a viscous fluid flow past a circular cylinder. − J. Phys. Soc. Jpn, vol.21, pp.2055-2062.
[14] Takami H. and Keller H.B. (1969): Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder. − Phys. Fluids Suppl., vol.12, pp.51-56.
[15] Dennis S.C.R. and Chang G.Z. (1970): Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. − J. Fluid Mech., vol.42, pp.471-489.
[16] Fornberg B. (1980): A numerical study of steady viscous flow past a circular cylinder. − J. Fluid Mech., vol.98, pp.819-855.
[17] Fornberg B. (1985): Steady viscous flow past a circular cylinder up to Reynolds number 600. − J. Comput. Phys., vol.61, pp.297-320.
[18] Henderson R.D. (1995): Details of the drag curve near the onset of vortex shedding. − Phys. Fluids, vol.7, pp.2102-2104.
[19] Chen J.H. (2000): Laminar separation of flow past a circular cylinder between two parallel plates. − Proc. Natl Sci. Counc. ROC A, vol.24, pp.341-351.
[20] Wu M.H., Wen C.Y., Yen R.H., Weng M.C. and Wang A.B. (2004): Experimental and numerical study of the separation angle for flow around a circular cylinder at low Reynolds number. − J. Fluid Mech., vol.515, pp.233-260.
[21] Sen S., Mittal S. and Biswas G. (2009): Steady separated flow past a circular cylinder at low Reynolds numbers. − J. Fluid Mech., vol.620, pp.89-119.
[22] Wei D.J., Yoon H.S. and Jung J.H. (2016): Characteristics of aerodynamic forces exerted on a twisted cylinder at a low Reynolds number of 100. − Comput. Fluids, vol.136, pp.456-466.
[23] Okajima A. (1982): Strouhal numbers of rectangular cylinders. − J. Fluid Mech., vol.123, pp.379-398.
[24] Okajima A., Nagashisa T. and Rokugoh A. (1990): A numerical analysis of flow around rectangular cylinders. − JSME Int. Series II, vol.33, pp.702-717.
[25] Mukhopadhaya A., Biswas, G. and Sundararajan T. (1992): Numerical investigation of confined wakes behind a square cylinder in a channel. − Int. J. Numer. Methods Fluids, vol.14, pp.1437-1484.
[26] Sohankar A., Norberg C. and Davidson L. (1998): Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition. − Int. J. Numer. Methods Fluids, vol.26. pp.39-56.
[27] Breuer M., Bernsdorf J., Zeiser T. and Durst F. (2000): Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume. − Int. J. Heat Fluid Flow, vol.21, pp.186-196.
[28] Gupta A.K., Sharma A., Chhabra R.P. and Eswaran V. (2003): Two-dimensional steady flow of a power-law fluid past a square cylinder in a plane channel: momentum and heat-transfer characteristics. − Ind. Eng. Chem. Res., vol.42, pp.5674-5686.
[29] Sen S., Mittal S. and Biswas, G. (2010): Flow past a square cylinder at low Reynolds numbers. − Int. J. Numer. Methods Fluids, vol.67. pp.1160-1174.
[30] Mahir N. (2017): Three dimensional heat transfer from a square cylinder at low Reynolds numbers. − Int. J. Thermal Sciences, vol.119, pp.37-50.
[31] Patanker S.V. (1980): Numerical Heat Transfer and Fluid Flow. − New York: McGraw-Hill.
[32] Demirdzic I. and Peric M. (1990): Finite volume method for prediction of fluid flow in arbitrary shaped domains with moving boundary. − Int. J. Numer. Methods Fluids, vol.10, pp.771-790.
[33] Patanker S.V. and Spalding D.B. (1972): A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. − Int. J. Heat Mass Transfer, vol.15, pp.1787-1806.
[34] Stone H.L. (1968): Iterative solution of implicit approximations of multidimensional partial differential equations. − SIAM. J. Numerical Analysis, vol.5, pp.530-558.
[35] Rhie C.M. (1981): A Numerical Study of the Flow Past an Isolated Airfoil with Separation. − PhD Thesis, Dept. of Mechanical and Industrial Engineering. University of Illinois at Urbana-Champaign.
[36] Peric M. (1990): Analysis of pressure-velocity coupling on non-orthogonal grids. − Numer. Heat Transfer, Part B, vol.17, pp.63-82.
[37] Homann F. (1936): Einfluss grosser zahigkeit bei stromung um zylinder. − Forsch. Ing. Wes., vol.7, pp.1-10.
[38] Sobey I.J. (2000): Introduction to Interactive Boundary Layer Theory. − Oxford University Press.
[39] Smith F.T. (1981): Comparisons and comments concerning recent calculations for flow past a circular cylinder. − J. Fluid Mech., vol.113, pp.407-410.
[40] Paliwal B., Sharma A., Chhabra R.P. and Eswaran V. (2003): Power law fluid flow past a square cylinder: momentum and heat transfer characteristics. − Chem. Eng. Sci., vol.58, pp.5315-5329.
[41] Hamielec A.E. and Raal J.D. (1969): Numerical studies of viscous flow around circular cylinders. − Phys. Fluids., vol.12, pp.11-17.
[42] Posdziech O. and Grundmann R. (2007): A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder. − J. Fluids Struct., vol.23, pp.479-499.