This work is devoted to the development and application of the numerical technique suitable for solution of the free-boundary problems, i.e. those in which the shape of the boundary should be determined as a part of the solution. The technique is based on a finite-difference solution of the equations of the problem on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The same orthogonal mapping approach may also be used to construct orthogonal coordinates fitted to boundaries of known but complicated shapes.
The technique is applied to two classical problems of fluid mechanics -- deformation of a gas bubble rising through a quiescent fluid due to buoyancy, and deformation of a gas bubble in a uniaxial extensional flow. For the rising bubble, the shapes and flow fields are computed for Reynolds numbers 1 ≤ R ≤ 200 and Weber numbers up to 20 at the lower Reynolds numbers and up to 10 at Reynolds numbers 50, 100 and 200. The most interesting results of this part are those demonstrating the phenomenon of flow separation at a smooth free surface. This phenomenon does not appear to have been theoretically predicted before, in spite of its importance for understanding the mechanics of free-surface flows.
In the case of a bubble in a uniaxial extensional flow, the computations show that at Reynolds numbers of order 10 and higher the deformation of a bubble proceeds in a way qualitatively different from the low Reynolds number regime studied previously; the bubble bursts at a relatively early stage of deformation never reaching the highly elongated shapes observed and predicted at low Reynolds numbers. It is shown also that for this problem the solution at Reynolds number of order 100 is already quite close to the potential flow solution which can be easily obtained using the present technique.
