A model of flow separation at a free surface

Abstract

Flow separation can be observed (1) at the leading edge of a spilling breaker or ‘white-cap’, (2) at the lower edge of a tidal bore or hydraulic jump and (3) upstream of an obstacle abutting a steady free-surface flow. At the point of flow separation there is a discontinuity in the slope of the free surface. The flow upstream of this point is relatively smooth; the flow downstream of the discontinuity is turbulent.In this note, a local solution for the flow in the neighbourhood of the discontinuity is derived. The turbulence is represented by a constant eddy viscosity N, and the tangential stress across the interface between the laminar and turbulent zones is expressed in terms of a drag coefficient C. It is shown that the inclinations of the free surface of the two sides of the discontinuity depend on C only, and are independent of N and g. As C increases from zero to large values, so the inclination of the free surface in the turbulent zone increases from 10° 54′ to 30°. In the laminar zone the inclination of the free surface simultaneously decreases from 10° 54′ to 0°, the densities in the two zones being assumed equal.Owing to the possible entrainment of air at the separation point, the effective density ρ′ in the turbulent zone may be less than the density ρ in the laminar zone. When these densities are allowed to be different it is found that the possible flows are of two distinct types. Flows of the first type, called ‘quasi-static’, are contiguous to a state of rest. Flows of the second type, called ‘dynamic’, are contiguous with the frictional flows described above, for which ρ′ = ρ At a given positive value of C there exists generally only one quasi-static solution. There is also just one dynamic solution provided ρ′/ρ > 0·50012. On the other hand, if ρ′/ρ < 0·5 there may be either two or no dynamic flows, depending on the value of C; and when 0·5 < ρ′/ρ > 0·50012 there may be three such flows.The inclination of the free surface is studied as a function of C and ρ′/ρ.