Abstract
Free boundaries are a characteristic feature in many problems in ideal hydrodynamics, e.g., in jet problems. The term "free boundary" refers to parts of the boundary of the flow region which are not known in advance and on which two boundary conditions are specified: kinematic and dynamic. Free-boundary problems are usually nonlinear, difficult to analyze, and so far have been little studied, with few exceptions, despite the voluminous literature on this subject. A survey of the classical results on these problems can be found, e.g., in the monographs [1-3]. In this paper, we consider two types of ideal-fluid plane flow with internal free boundaries in a channel of a complex shape. These flows are described by elliptical equations for the stream functions. We study flow in Lavrent'ev's modified scheme [4], which joins potential and turbulent flows with an unknown separation boundary. Vorticity is defined up to a constant, which is determined from the total circulation on the boundary of the turbulent region. A numerical algorithm is constructed, based on the fictitious region method. The flow of the two-layer fluid is modeled by the Laplace equation for the stream functions. The specific feature of the problem is the presence of nonhomogeneous matching conditions on the unknown boundary. The generalized formulation of the problem and the fictitious region method produce an efficient algorithm and numerical through schemes. The nonlinear problems are solved using an iterative process of successive approximations by the nonlinearity (the free boundary). 1. Joining Problem. Consider the problem of joining potential and turbulent flows with an unknown separation boundary (Lavrent'ev's scheme [4]). The flow region is D = D o O D1, with potential flow in D O and turbulent flow in D 1. The stream function ~p is such that ~(x) = 0 for x ~ S = OD o tA OD1,
Published Version
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