Flow with a free surface

If a liquid is flowing with a free surface exposed to the surroundings, the pressure at the liquid surface will everywhere be constant and equal to atmospheric pressure. Flow will take place feerefore only as a result of tihe action of the gravitational force, and the surface level will necessarily fall in the direction of flow. 

Two cases are considered. The first, the laminar flow of a thin film down an inclined surface, is importmit in the heat transfer from a condensing vapour where the main resistance to transfer lies in the condensate film, as discussed in Chapter 9 (Section 9.6.1). The second is the flow in open channels which are frequently used for transporting liquids down a slope on an industrial site. 

In any liquid flowing down a surface, a velocity profile is established with the velocity increasing from zero at the surface itself to a maximum where it is in contact with the surrounding atmosphere. The velocity distribution may be obtained in a manner similar to that used in connection witii pipe flow, but noting that the driving force is that due to gravity rather than a pressure gradient. 

If the drag force of the atmosphere is negligible, the retarding force for laminar flow is attributable to the viscous drag in the liquid at flie distance y from the surface given by  

Since there will normally be no slip between the liquid and the surface, then Ux 

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A general equation for frictional resistance in a pressure conduit was developed. The same reasoning may now be applied to uniform flow with a free surface. Consider a sloped channel of water flowing over a constant slope of angle 0, which shows the short reach of length L between stations 1 and 2 of a channel in uniform flow with area A of the water section. As the flow is neither accelerating nor decelerating,... 

Flow with a free surface is discussed in Section 3.6 and the particular case of laminar flow down an inclined surface in Section 3.6.1. For a flow of liquid of depth 8, width w and density p down a surface inclined at an angle 0 to the horizontal, a force balance in the x direction (parallel to the surface) may be written. The weight of fluid flowing down the plane at a distance y from the free surface is balanced by the shear stress at the plane. For unit width and unit height ... 

Froude w inertia force flow with a free surface,... 

Haque et al [35] simulated turbulent flows with a free-surface in unbaffled agitated vessels using three turbulence models (i.e., the Reynolds stress, standard k-e and k-ix) SST models) as implemented in CFX. In this case it appears that the SST model did perform better than the standard k-e model, whereas some features of the flow structure and the mean velocity profiles were better predicted by the Reynolds stress model. 

Orr E.M. Jr 1976. Numerical Simulation of Viscous Flow with a Free Surface (by Galerkin s Method with Finite Element Basis Eunctions). [LES]... 

Example Gravity Driven Flow with a Free Surface... 

Harlow FH, Welsh JE (1965) Numerical calculation of time dependent viscous incompressible flow with a free surface. Physics of Fluids 8 2182-2189... 

D. G. Dommermuth and D. K. P. Yue, Numerical simulations of nonlinear axisym-metric flows with a free surface, J. Fluid Mech. 178, 195-219 (1987). 

Wilson, K. C. 1980. Analysis of slurry flows with a free surface. Paper C4 read at Hydrotransport 7, Sendai, Japan. Cranfield, UK BHRA Group, pp 123-132. 

Electrospinning (ES) represents a facile and versatile tool for manufacturing nanofibers. The ES process is involved with the polymer flow with a free surface after appearing from a... 

Shao, S., Lo, E. Y. M. (2003). Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources, 26, 787-800. 

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