Abstract
Steady, trans-critical flow of a two-fluid system over a semi-circular cylinder on the bottom of a channel is considered. Each fluid is assumed to be inviscid and incompressible and to flow irrotationally, but the fluids have different densities, so that one flows on top of the other. Consequently, a sharp interface exists between the fluids, in addition to a free surface at the top of the upper fluid. Trans-critical flow is investigated, in which waves are absent from the system, but the upstream and downstream fluid depths differ in each fluid layer. The problem is formulated using conformal mapping and a system of three integrodifferential equations, and solved numerically with the aid of Newton's method. The free-surface shape and that of the interface are obtained along with the Froude numbers in each fluid layer. Results of computation are presented and discussed.
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References
J.M. Aitchison, A variable finite element method for the calculation of flow over a weir, Rutherford Laboratory report No. RL-79-069 (1979).
L. Armi, The hydraulics of two flowing layers with different densities, J. Fluid Mech. 163 (1986) 27–58.
L. Armi and D.M. Farmer, Maximal two-layer exchange through a contraction with barotropic net flow, J. Fluid Mech. 164 (1986) 27–51.
G.S. Benton, The occurrence of critical flow and hydraulic jumps in a multi-layered fluid system, J. Met. 11 (1954) 139–150.
P. Bettess and J.A. Bettess, Analysis of free surface flows using isoparametric finite elements, Int. J. Num. Meth. Eng. 19 (1983) 1675–1689.
D.M. Farmer and L. Armi, Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic flow, J. Fluid Mech. 164 (1986) 53–76.
L.K. Forbes, On the effects of non-linearity in free-surface flow about a submerged point vortex, J. Eng. Math. 19 (1985) 139–155.
L.K. Forbes, A numerical method for non-linear flow about a submerged hydrofoil, J. Eng. Math. 19 (1985) 329–339.
L.K. Forbes, Critical free-surface flow over a semi-circular obstruction, J. Eng. Math. 22 (1988) 3–13.
L.K. Forbes and L.W. Schwartz, Free-surface flow over a semicricular obstruction, J. Fluid Mech. 114 (1982) 299–314.
F.M. Henderson, Open Channel Flow, Macmillan and Co., New York (1966).
W.K. Melville and K.R. Helfrich, Transcritical two-layer flow over topography, J. Fluid Mech. 178 (1987) 31–52.
V.J. Monacella, On ignoring the singularity in the numerical evaluation of Cauchy Principal Value integrals, Hydromechanics Laboratory research and development report 2356, David Taylor Model Basin, Washington D.C. (1967).
P.M. Naghdi and L. Vongsarnpigoon, The downstream flow beyond an obstacle, J. Fluid Mech. 162 (1986) 223–236.
N.S. Sivakumaran, T. Tingsanchali and R.J. Hosking, Steady shallow flow over cured beds, J. Fluid Mech. 128 (1983) 469–487.
J.-M. Vanden-Broeck, Free-surface flow over an obstruction in a channel, Phys. Fluids 30 (1987) 2315–2317.
J.-M. Vanden-Broeck and J.B. Keller, Weir flows, J. Fluid Mech. 176 (1987) 283–293.
I.R. Wood and K.K. Lai, Flow of layered fluid overbroad crested weir, J. Hyd. Div. ASCE 98 (1972) 87–104.
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Forbes, L.K. Two-layer critical flow over a semi-circular obstruction. J Eng Math 23, 325–342 (1989). https://doi.org/10.1007/BF00128906
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DOI: https://doi.org/10.1007/BF00128906