Simulation of oil-water mixturesMovie 1 shows a finite-volume, level_set simulation of two-phase flow at Reynolds number 100 in two dimensions. The geometry is a simple square with no slip applied on each wall, except the top wall which is driven from left to right (the classic lid-driven cavity flow). The liquids are identical except that we have made them immiscible. We used a finite volume method on a fully staggered grid to solve the Navier-Stokes equations. The interface between the liquids is advected (passively) by the local fluid velocity and this is solved by a level set method. However, in this specific case, the level set method becomes really just the advection of a passive scalar which demarcates the blue and red regions. I.e., think of a tracer dye that does not diffuse. Note that surface tension is not accounted for. The grid resolution is 50 X 50. A special reinitialisation method is applied to ensure that the area is preserved (and hence the mass). In absence of this reinitialisation, the level set method would not be conservative (as in Movie 1b). The white dots are merely passive marker particles which we advected with the local fluid velocity. They serve to aid in visualising the flow development, nothing more. Movie 2 shows a simulation using the same numerical methods but, now with gravity acting downwards and the fluid densities and viscosities being unequal. (They are still immiscible and there is no surface tension). Notice that the mass of the red fluid is not conserved in the simulation. This is a known deficiency of the level set methods. So far, surface tension has been neglected. Movie 3 shows a simulation of square oil "bubble" in water (with gravity neglected). The surface tension will force the bubble back to the usual circular shape. Simulation of air-water mixturesThe finite-volume, level-set method can also be extended to flows with large density differences, e.g., an air-water mixture. Movie 4 shows an animation of a dambreak problem in a closed square container. Movie 5 shows a 2D simulation of air and water in a 2D channel inclined at 10 degrees with gravity acting downwards. The water is coloured blue and the air white. The red dots are passive tracer particles to aid visualisation. The channel is 10 meters long and 1 meter high. Surface tension effects are negligible at such long length scales and were therefore ignored, but viscous effects were included. The flow is also forced to maintain a constant average flow rate. Periodic boundary conditions are applied in the flow direction. Free slip conditions are applied at the lower and upper channel walls. This simulation was started with an initial fluid distribution where a layer of air sits above the main body of water. A slight distortion of the air-water interface was also introduced in order to promote instabilities which develop into ripples on the interface. The mean flow rate is 1 meter per second which is maintained by a time-dependent forcing term. Finally, the total simulation time is 10 seconds (the frame rate is not scaled properly though). Movie 6a and 6b shows a second air-water simulation. The simulation uses the same geometry and boundary conditions as in Movie 5, except that we started the flow with an initial fluid distribution consisting of a rectangular air pocket situated at the upper channel wall. Also, the mean flow rate is 2 meters per second in order to accelerate the dynamics. Once again the simulation was run up to 10 seconds. The simulations show the formation of gravity waves at early times, the elongation of the bubble and its eventual distortion to the point of shedding bubbles. | 