Abstract
High-Reynolds-number (Re) flow containing closed streamlines (Prandtl-Batchelor flows), within a region enclosed by a smooth boundary at which the boundary conditions are discontinuous, is considered. In spite of the need for local analysis to account fully for flow at points of discontinuity, asymptotic analysis for Re ≪ 1 indicates that the resulting mathematical problem for determining the uniform vorticity ω0) in these situations, requiring the solution of periodic boundary-layer equations, is in essence the same as that for a flow with continuous boundary data. Extensions are proposed to earlier work [3] to enable ω0 to be computed numerically; these require coordinate transformations for the boundary-layer variables at singularities, as well as a two-zone numerical integration scheme. The ideas are demonstrated numerically for the classical circular sleeve.
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References
G. K. Batchelor, On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech.1 (1956) 177-190.
N. Riley, High Reynolds number flows with closed streamlines. J. Eng. Math.15 (1981) 15-27.
M. Vynnycky, On the uniform vorticity in a high Reynolds number flow. J. Eng. Math.28 (1994) 129-144.
E. W. Haddon and N. Riley, On flows with closed streamlines. J. Eng. Math.19 (1985) 233-246.
W. H. Lyne, Unsteady viscous flow in a curved pipe. J. Fluid Mech.45 (1970) 13-31.
S. I. Chernyshenko, An approximate method of determining the vorticity in the separation region as the viscosity tends to zero. Fluid Dyn.17 (1982) 1-11.
S. I. Chernyshenko, The calculation of separated flows of low viscosity liquids using the Batchelor model. Library translation 2133, RAE (1985).
S. I. Chernyshenko, The asymptotic form of the steady separated flow past a body at large Reynolds number. Appl. Math. Mech.52 (1988) 746-753.
S. I. Chernyshenko and I. P. Castro, High Reynolds-number asymptotics of the steady flow through a row of bluff bodies. J. Fluid Mech.257 (1996) 421-449.
S. I. Chernyshenko and I. P. Castro, High Reynolds-number weakly stratified flow past an obstacle. J. Fluid Mech.317 (1996) 155-178.
F. T. Smith, A structure for laminar flow pasta bluff body at high Reynolds number. J. Fluid Mech.155 (1985) 175-191.
D. H. Peregrine, A note on the steady high Reynolds-number flow about a circular cylinder. J. Fluid Mech. 157 (1985) 493-500.
F. T. Smith, Concerning inviscid solutions for large-scale separated flows. J. Eng. Math.20 (1986) 271-292.
R. McLachlan, A steady separated viscous corner flow, J. Fluid Mech.231 (1991) 1-34.
G. Giannakidis, Prandtl-Batchelor flow in a channel. Phys. Fluids A5 (1993) 863-867.
C. Turfus, Prandtl-Batchelor flow past a flat plate at normal incidence in a channel - inviscid analysis. J. Fluid Mech.249 (1993) 59-72.
G. Giannakidis, Prandtl-Batchelor flow on a circular cylinder and on aerofoil sections. Aeronaut. J.100 (1996) 15-26.
M. Vynnycky and K. Kanev, Coupled Batchelor flows in a confined cavity. J. Fluid Mech.319 (1996) 305-322.
A. V. Bunyakin, S. I. Chernyshenko and G.Y. Stepanov, Inviscid Batchelor-model flow past an airfoil with a vortex trapped in a cavity. J. Fluid Mech.323 (1996) 367-376.
S. Goldstein, Concerning some solutions of the boundary-layer equations in hydrodynamics. Proc. Camb. Phil. Soc.26 (1930) 1-30.
A. J. Van de Vooren and D. Djikstra, The Navier-Stokes solution for laminar flow past a semi-infinite plate. J. Eng. Math.4 (1970) 9-27.
F. T. Smith, Boundary-layer flow near a discontinuity in wall conditions. J. Inst. Maths and its Applics.13 (1974) 127-145.
A. E. P. Veldman, A new calculation of the wake of a plate. J. Eng. Math.9 (1975) 65-70.
T. Cebeci, F. Thiele, P. G. Williams and K. Stewartson, On the calculation of symmetric wakes I. Two-dimensional flows. Num. Heat Transf.2 (1979) 35-60.
T. Cebeci and P. Bradshaw, Momentum Transfer in Boundary Layers. Washington: Hemisphere Publishing Corporation (1977) 391pp.
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge University Press (1989) 818pp.
E.W. Haddon and N. Riley, A note on the mean circulation in standing waves. Wave Motion5 (1983) 43-48.
D. W. Moore, P. G. Saffman and S. Tanveer, The calculation of some Batchelor flows: The Sadovskii vortex and rotational corner flow. Phys. Fluids31 (1988) 978-990.
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Vynnycky, M. Concerning closed-streamline flows with discontinuous boundary conditions. Journal of Engineering Mathematics 33, 141–156 (1998). https://doi.org/10.1023/A:1004204527294
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DOI: https://doi.org/10.1023/A:1004204527294