Abstract: The variational principle in terms of stream function ψ for free surface gravity flow is discussed by the formulation of first-order variation in a variable domain. Because of different transversal conditions adopted, there are four forms of variational principle in terms of ψ.A n air-gilled cavity flow with given discharge and total energy is then analysed by finite element method. At the end of the cavity, the free stream line is tangent to a short fictitious plate of given length, which joins the fixed boundary at on angle to be determined. The condition that the free stream line should be tangent to the fixed boundary at the point of separation makes the solution unique.Finally curves giving the cavity length as a function of the Fraude number, cavity pressure and channel bottom slope are presented.

Key words: Ehresmann space, basic equation, Cauchy problem