Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-18T22:25:56.417Z Has data issue: false hasContentIssue false
  • English
  • Français

A study of two-dimensional flow past regular polygons via conformal mapping

Published online by Cambridge University Press:  01 June 2009

ZHONG WEI TIAN  and
ZI NIU WU
ZHONG WEI TIAN
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
ZI NIU WU*
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study two-dimensional flow around regular polygons with an arbitrary but even number of edges N and one apex pointing to the free stream, with comparison to circular-cylinder flow. Both inviscid flow and low-Reynolds-number viscous flow are addressed. For inviscid flow, we obtained the exact solution for pure potential flow through Schwarz–Christoffel transformation, with the emphasis on the role of edge number, N, on the flow details. We also studied the behaviour, stationary lines and stability of vortex pair and found new stationary lines compared to circular cylinder. For viscous flow we derived the equation of stream function in the mapped (circle) domain, based on which approximate expressions for the critical Reynolds numbers and Strouhal number, as functions of the edge number, are obtained. The Reynolds number is based on the diameter of the circumscribed circle. For the steady flow, the first critical Reynolds number is a monotonically decreasing function of N, while N → ∞ corresponds to that for circular cylinder. The bifurcation point is ahead of the bifurcation point for circular cylinder. For unsteady flow, the critical Reynolds number for vortex shedding and the Strouhal number are both monotonically decreasing functions of N.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 628 , 10 June 2009 , pp. 121 - 154
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abbassi, H., Turki, S. & Nasrallah, S. B. 2002 Channel flow past bluff-body: outlet boundary condition, vortex shedding and effects of buoyancy. Comput. Mech. 28, 10–16.CrossRefGoogle Scholar
Barkley, D. & Henderson, R. D. 1996 Three dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Breuer, M., Bernsdorf, J., Zeiser, T. & Durst, F. 2000 Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume. Intl J. Heat Fluid Flow 21, 186–196.CrossRefGoogle Scholar
Brøons, M., Jakobsen, B. & Niss, K. 2007 Streamline topology in the near wake of a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 584, 23–43.CrossRefGoogle Scholar
Cai, J., Liu, F. & Luo, S. J. 2003 Stability of symmetric vortices in two dimensions and over three-dimensional slender conical bodies. J. Fluid Mech. 480, 65–94.CrossRefGoogle Scholar
Clements, R. R. 1973 An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57, 321–336.CrossRefGoogle Scholar
Coutanceau, M. & Daefaye, J. R. 1991 Circular cylinder wake configurations: a flow visualization survey. Appl. Mech. Rev. 44 (6), 255–305.CrossRefGoogle Scholar
De, A. K. & Dalal, A. 2006 Numerical simulation of unconfined flow past a triangular cylinder. Intl J. Numer. Mech. Fluids 52, 801–821.Google Scholar
Dennis, S. C. R. & Chang, G. Z. 1970 Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech. 42, 379–399.CrossRefGoogle Scholar
Dimopoulos, H. G. & Hanratty, T. J. 1968 Velocity gradients at the wall for flow around a cylinder for Reynolds numbers between 60 and 360. J. Fluid Mech. 33, 303–319.CrossRefGoogle Scholar
Dušek, J., Gal, P. L. & Fraunié, P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59–80.CrossRefGoogle Scholar
Elcrat, A., Fornberg, B., Horn, M. & Miller, K 2000 Some steady vortex flows past a circular cylinder. J. Fluid Mech. 409, 13–27.CrossRefGoogle Scholar
Elcrat, A. R. & Trefethen, L. N. 1986 Classical free-streamline flow over a polygonal obstacle. J. Comput. Appl. Math. 14, 251–265.CrossRefGoogle Scholar
Fey, U., König, M. & Eckelmann, H. 1998 A new Strouhal–Reynolds-number relationship for the circular cylinder in the range 47 < re < 2 × 105. Phys. Fluids 10 (7), 1547–1549.CrossRefGoogle Scholar
Föppl, L. 1913 Wirbelbewegung hinter einem kreiszylinder [Vortex motion behind a circular cylinder]. Sitzb. d. k. baeyr, Akad. d. Wiss. 1, 1–17.Google Scholar
Franke, R. 1991 Numerische berechnung der instationären wirbelablösung hinter zylindrischen körpern. PhD thesis, University of Karlsruhe.Google Scholar
Grove, A. S., Shair, F. H., Petersen, E. E. & Acrivos, A. 1964 An experimental investigation of the steady separated flow past a circular cylinder. J. Fluid Mech. 19, 60–81.CrossRefGoogle Scholar
Henderson, R. D. 1995 Details of the drag curve near the onset of vortex shedding. Phys. Fluids 7 (9), 2102–2104.CrossRefGoogle Scholar
Homann, F. 1936 Einfluss grsser zähigkeit bei strmung um zylinder. Forsch. Ing. Wes. 7, 1–9.CrossRefGoogle Scholar
Igarashi, T. 1997 Drag reduction of a square prism by the flow control using a small rod. J. Wind Engng Indus. Aerodyn. 69–71, 141–153.CrossRefGoogle Scholar
Igarashi, T. & Ito, S. 1993 Drag reduction of a square prism. Part 1. Flow control around a square prism using a small vortex shedder. Trans. JSME 59 (568), 3701–3707.CrossRefGoogle Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.CrossRefGoogle Scholar
Kiya, M., Saskai, K. & Arie, M. 1982 Discrete vortex simulation of a turbulent separation bubble. J. Fluid Mech. 120, 219–244.CrossRefGoogle Scholar
Klekar, K. M. & Patankar, S. V. 1992 Numerical prediction of vortex shedding behind square cylinders. Intl J. Numer. Mech. Fluids 14, 327–341.CrossRefGoogle Scholar
Lin, C. C. 1941 a On the motion of vortices in two dimensions – I Existence of the Kirchhoff Routh function. Proc. Natl Acad. Sci. 27 (12), 570–575.CrossRefGoogle ScholarPubMed
Lin, C. C. 1941 b On the motion of vortices in two dimensions – II Some further investigations on the Kirchhoff Routh function. Proc. Natl Acad. Sci. 27 (12), 575–577.CrossRefGoogle ScholarPubMed
Luo, S. C., Chew, Y. T. & Ng, Y. T. 2003 Characteristics of square wake transition flows. Phys. Fluids 15 (9), 2549–2559.CrossRefGoogle Scholar
Newton, P. K. 2000 The N-Vortex Problem. Springer.Google Scholar
Noack, B. R. & Eckelmann, H. 1994 a A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.CrossRefGoogle Scholar
Noack, B. R. & Eckelmann, H. 1994 b A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids 6, 124–143.CrossRefGoogle Scholar
Okajima, A. 1982 Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398.CrossRefGoogle Scholar
Posdziech, O. & Grundmann, R. 2001 Numerical simulation of the flow around an infinitely long circular cylinder in the transition regime. Theoret. Comput. Fluid Dynamics 15, 121–141.Google Scholar
Protas, B. 2007 Center manifold analysis of a point-vortex model of vortex shedding with control. Physica D 228 (2), 179–187.CrossRefGoogle Scholar
Robichaux, J., Balachandar, S. & Vanka, S. P. 1999 Three-dimensional Floquet instability of the wake of square cylinder. Phys. Fluids 11, 560–578.CrossRefGoogle Scholar
Routh, E. J. 1881 Some applications of conjugate functions. Proc. Lond. Math. Soc. 12, 73–89.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saha, A. K., Biswas, G. & Muralidha, K. 2003 Three dimensional study of flow past a square cylinder at low Reynolds numbers. Intl J. Heat Fluid Flow 24, 54–66.CrossRefGoogle Scholar
Sakamoto, H., Tan, K., Takeuchi, N. & Haniu, H. 1997 Suppression of fluid forces acting on a square prism by passive control. ASME J. Fluids Engng 119, 506–511.CrossRefGoogle Scholar
Sandy, S. C. 1970 Evaluation of the Schwarz–Christoffel mapping function for special polygons. SIAM J. Appl. Math. 18 (4), 815–817.CrossRefGoogle Scholar
Sarpkaya, T. 1975 An inviscid model of two dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate. J. Fluid Mech. 68, 109–128.CrossRefGoogle Scholar
Schumm, M., Berger, E. & Monkewitz, P. A. 1994 Self-excited oscillations in the wake of two-dimensional bluff bodies and therir control. J. Fluid Mech. 271, 17–53.CrossRefGoogle Scholar
Sharma, A. & Eswaran, V. 2004 Heat and fluid flow across a square cylinder in the two-dimensional laminar flow regime. Numer. Heat Transfer A 45, 247–269.CrossRefGoogle Scholar
Shashikanth, B. N., Marsden, J. E., Burdick, J. W. & Kelly, S. D. 2002 The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with n point vortices. Phys. Fluids 14 (3), 1214–1227.CrossRefGoogle Scholar
Skews, B. W. 1991 Autorotation of many-sided bodies in an airstream. Nature 352, 512–513.CrossRefGoogle Scholar
Skews, B. W. 1998 Autorotation of polygonal prisms with an upstream vane. J. Wind Engng Indus. Aerodyn. 73, 145–158.CrossRefGoogle Scholar
Sohankar, A. 2007 Hopf bifurcation, vortex shedding and near wake study of a heated cylinder in cross flow. Iranian J. Sci. Technol. 31 (B1), 31–47.Google Scholar
Sohankar, A., Norberg, C. & Davidson, L. 1999 Simulation of three dimensional flow around a square cylinder at moderate Reynolds numbers. Phys. Fluids 11, 288–306.CrossRefGoogle Scholar
Taneda, S. 1956 Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Jpn 11, 347–384.Google Scholar
Tang, S. & Aubry, N. 1997 On the symmetry breaking instability leading to vortex shedding. Phys. Fluids 9 (9), 2550–2561.CrossRefGoogle Scholar
Thom, A. 1933 The flow past circular cylinders at low speeds. Proc. R. Soc. Lond. A 141, 651–669.Google Scholar
Thompson, M. C. & Gal, P. L. 2004 The stuart – landau model applied to wake transition revisited. Eur. J. Mech. B 23, 219–228.CrossRefGoogle Scholar
Tian, X. S. 1996 Wind tunnel test investigation and engineering application of 24 angular polygon section. Acta Aerodyn. Sin. 14 (4), 379–386.Google Scholar
Wesfreid, J. E., Goujon-Durand, S. & Zielinska, B. J. A. 1996 Global mode behavior of the streamwise velocity in wakes. J. Phys. II France 6, 1343–1357.Google Scholar
Williamson, C. H. K. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.CrossRefGoogle Scholar
Wu, M. H., Wen, C. Y., Yen, R. H., Weng, M. C. & 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. 515, 233–260.CrossRefGoogle Scholar
Zannetti, L. 2006 Vortex equilibrium in flow past bluff bodies. J. Fluid Mech. 562, 151–171.CrossRefGoogle Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21, 155–165.CrossRefGoogle Scholar
Zhang, H. Q., Fey, U., Noack, B. R., König, M. & Eckelmann, H. 1995 On the transition of the cylinder wake. Phys. Fluids 7 (4), 779–794.CrossRefGoogle Scholar
Zhou, L., Cheng, M. & Hung, K. C. 2005 Suppression of fluid force on a square cylinder by flow control. J. Fluids Struct. 21, 151–167.CrossRefGoogle Scholar
Zielinska, B. J. A., Durand, S. G., Dušek, J. & Wesfreid, J. E. 1997 Strongly nonlinear effect in unstable wakes. Phys. Rev. Lett. 79, 3893–3896.CrossRefGoogle Scholar
Zielinska, B. J. A. & Wesfreid, J. E. 1995 On the spatial structure of global modes in wake flow. Phys. Fluids 7, 1418–1424.CrossRefGoogle Scholar