Abstract
The problem of viscous prograde (eastward) and retrograde (westward) flow past a cylindrical obstacle on a β-plane is considered. The barotropic vorticity equation is solved using a numerical method that combines finite difference and spectral methods. A modified version of the β-plane approximation is proposed to avoid computational difficulties associated with the traditional β-plane approximation. Numerical results are presented and discussed for flow past a circular cylinder at selected Reynolds numbers (Re) and non-dimensional β-parameters (\({\hat{\beta}}\)) as well as for flow past an elliptic cylinder of a fixed aspect ratio (r = 0.2) at inclination angles of ±15°, 90° and selected Re and \({\hat{\beta}}\) . In prograde flows, it is found that the β-effect acts to suppress boundary-layer separation and to excite a standing Rossby lee wavetrain, as observed in previous works. In retrograde flows, the boundary-layer separation region is elongated and westward propagating Rossby waves are excited.
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References
Badr H., Dennis S.: Time dependent viscous flow past an impulsively started rotating and translating circular cylinder. J. Fluid Mech. 158, 447–488 (1985)
Badr H., Dennis S., Kocabiyik S.: Numerical simulation of unsteady flow over an elliptic cylinder at different orientations. Int. J. Numer. Methods Fluids 37, 905–934 (2001)
Boyer D., Davies A.: Flow past a circular cylinder on a beta-plane. Philos. Trans. R. Soc. Lond. A 306, 533–556 (1982)
Bryan K.: A numerical investigation of a nonlinear model of a wind-driven ocean. J. Atmos. Sci. 20, 594–606 (1963)
Dennis S., Chang G.Z.: Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech. 42, 471–489 (1970)
Dennis S., Quartapelle L.: Some uses of Green’s theorem in solving the Navier-Stokes equations. Int. J. Numer. Methods Fluids 9, 871–890 (1989)
Dennis S., Young P.: Steady flow past an elliptic cylinder inclined to the stream. J. Eng. Math. 47, 101–120 (2003)
Matsuura T.: The separation of flow past a circular cylinder on a β-plane. J. Oceanogr. Soc. Japan 42, 362–372 (1986)
McIntyre M., Palmer T.: Breaking planetary waves in the stratosphere. Nature 305, 593–600 (1983)
Merkine L.O.: Flow separation on a beta-plane. J. Fluid Mech. 99, 399–409 (1980)
Oseen, C.: Über die Stokes’sche Formel, umd Über eine verwandte Aufgabe in der Hydrodynamik. Ark. Math. Astron. Fys. 6(29) (1910)
O’Sullivan D., Hitchman M.: Inertial instability and Rossby wave breaking in a numerical model. J. Atmos. Sci. 49, 991–1002 (1992)
Page M., Johnson E.: Flow past cylindrical obstacles on a beta-plane. J. Fluid Mech. 221, 349–382 (1990)
Rhines P.: Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–433 (1975)
Steinmoeller, D.: Flow separation on the β-plane, M. Math thesis, University of Waterloo, Waterloo, Ontario (2009)
Tansley C., Marshall D.: Flow past a cylinder on a β-plane, with application to Gulf Stream separation and the Antarctic Circumpolar Current. J. Phys. Oceanogr. 31, 3274–3283 (2001)
Viúdez A.: A new interpretation of the beta term in the vorticity equation. J. Atmos. Sci. 60, 1866–1870 (2003)
White W.: A Rossby wake due to an island in an eastward current. J. Phys. Oceanogr. 1, 161–168 (1971)
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Steinmoeller, D.T., D’Alessio, S.J.D. & Poulin, F.J. Prograde and retrograde flow past cylindrical obstacles on a β-plane. Acta Mech 217, 157–176 (2011). https://doi.org/10.1007/s00707-010-0386-6
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DOI: https://doi.org/10.1007/s00707-010-0386-6