Subcritical free-surface flow caused by a line source in a fluid of finite depth G. C. Hocking L. K. Forbes Article Received: 25 January 1991 Accepted: 04 October 1991 DOI :
10.1007/BF00042763

Cite this article as: Hocking, G.C. & Forbes, L.K. J Eng Math (1992) 26: 455. doi:10.1007/BF00042763
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Abstract An integral equation is derived and solved numerically to compute the flow and the free surface shape generated when water flows from a line source into a fluid of finite depth. At very low values of the Froude number, stagnation point solutions are found to exist over a continuous range in the parameter space. For each value of the source submergence depth to free stream depth ratio, an upper bound on the existence of stagnation point solutions is found. These results are compared with existing known solutions. A second integral equation formulation is discussed which investigates the hypothesis that these upper bounds correspond to the formation of waves on the free surface. No waves are found, however, and the results of the first method are confirmed.

References 1.

A. Craya, Theoretical research on the flow of nonhomogeneous fluids.

La Houille Blanche 4 (1949) 44–55.

Google Scholar 2.

I.L. Collings, Two infinite Froude number cusped free surface flows due to a submerged line source or sink.

J. Aust. Math. Soc. Ser. B 28 (1986) 260–270.

Google Scholar 3.

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.

Google Scholar 4.

L.K. Forbes, A numerical method for non-linear flow about a submerged hydrofoil.

J. Eng. Math. 19 (1985) 320–339

Google Scholar 5.

L.K. Forbes and G.C. Hocking, Flow caused by a point sink in a fluid having a free surface.

J. Aust. Math. Soc. Ser. B 32 (1990) 231–249.

Google Scholar 6.

L.K. Forbes and G.C. Hocking. Suberitieal free-surface flow caused by a line source in a fluid of finite depth, Part 1.Mathematics Rept. 1991/01 . Department of Mathematics, University of Western Australia (1991).

7.

L.K. Forbes and G.C. Hocking, Flow induced by a line sink in a quiescent fluid with surface-tension effects.J. Aust. Math. Soc. ser. B (1992) in press.

8.

G.C. Hocking, Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom.

J. Aust. Math. Soc. Ser. B 26 (1985) 470–486.

Google Scholar 9.

G.C. Hocking, Infinite Froude number solutions to the problem of a submerged source or sink.

J. Aust. Math Soc. Ser. B 29 (1988) 401–409.

Google Scholar 10.

G.C. Hocking and L.K. Forbes, A note on the flow induced by a line sink beneath a free surface.

J. Aust. Math. Soc. Ser. B 32 (1991) 251–260.

Google Scholar 11.

G.C. Hocking, Withdrawal from a two layer fluid through a line sink.

J. Hydr. Engng., A.S.C.E. 117 (1991) 800–805.

Google Scholar 12.

G.C. Hocking, Critical withdrawal from a two-layer fluid through a line sink.

J. Eng. Math. 25 (1991) 1–11.

Google Scholar 13.

J. Imberger and P.F. Hamblin, Dynamics of lakes, reservoirs and cooling ponds

Ann. Rev. Fluid Mech. 14 (1982) 153–187.

Google Scholar 14.

J. Imberger and J.C. Patterson, Physical Limnology.

Adv. Appl. Mech. 27 (1989) 303–475.

Google Scholar 15.

A.C. King and M.I.G. Bloor, A note on the free surface induced by a submerged source at infinite Froude number.

J. Aust. Math. Soc. Ser. B 30 (1988) 147–156.

Google Scholar 16.

H. Mekias and J.M.Vanden Broeck, Supereritical free-surface flow with a stagnation point due to a submerged source.

Phys. Fluids A 1 (10) (1989) 1694–1697.

Google Scholar 17.

D.H. Peregrine, A line source beneath a free surface. Mathematics Research Center, Univ, Wisconsin Rept. 1248 (1972).

18.

E.O. Tuck and J.M.Vanden Broeck, A cusp-like free-surface flow due to a submerged source or sink.

J. Aust. Math. Soc. Ser B 25 (1984) 443–450.

Google Scholar 19.

J.M.Vanden Broeck and J.B. Keller, Free surface flow due to a sink.

J. Fluid Mech. 175 (1987) 109–117.

Google Scholar 20.

J.M.Vanden Broeck, L.W. Schwartz and E.O. Tuck, Divergent low-Froude number series expansion of non-linear free-surface flow problems.

Proc. Roy. Soc. London Ser. A 361 (1978) 207–224.

Google Scholar © Kluwer Academic Publishers 1992

Authors and Affiliations G. C. Hocking L. K. Forbes 1. Department of Mathematics University of Western Australia Nedlands Australia 2. Department of Mathematics University of Queesnland St. Lucia Australia