An adaptive multiphase hybrid level set moment-of-fluid method is developed to study the impact and solidification of water droplets on flat surfaces. The numerical simulations are validated by comparison to analytical results and experimental observations. The present simulations demonstrate the ability of the method to capture sharp solidification front, and handle contact line dynamics, and the simultaneous impact, merging and freezing of a drop. Parameter studies have been conducted, which show the influence of the Stefan number on the regularity of the shape of frozen droplets. Also, it is shown that impacting droplets with different sizes create ice shapes which are uniform near the impact point and become dissimilar away from it. In addition, surface wettability determines whether droplets freeze upon impact or bounce away.
A novel space-time discontinuous Galerkin (DG) spectral element method is presented to solve the one dimensional Stefan problem in an Eulerian coordinate system. This method employs the level set procedure to describe the time-evolving interface. To deal with the prior unknown interface, a backward transformation and a forward transformation are introduced in the space-time mesh. By combining an Eulerian description, i.e., a fixed frame of reference, with a Lagrangian description, i.e., a moving frame of reference, the issue of dealing with implicitly defined arbitrary shaped space-time elements is avoided. The backward transformation maps the unknown time-varying interface in the fixed frame of reference to a known stationary interface in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the space-time framework, are discretized by a DG spectral element method in each space-time slab. The forward transformation is used to update the level set function and then to project the solution in each phase back from the moving frame of reference to the fixed Eulerian grid. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. A Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that just a few iterations lead to convergence.
A space-time discontinuous Galerkin spectral element method is combined with two different approaches for treating problems with discontinuous solutions: (i) adding a space-time dependent artificial viscosity, and (ii) tracking the discontinuity with space-time spectral accuracy. A Picard iteration method is employed to solve nonlinear system of equations derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is demonstrated for the Burgers’ equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. The spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers’ equation obtained by the tracking method.
A novel block structured adaptive space-time spectral element and moment-of-fluid method is described for computing solutions to incompressible multi-phase/multi-material flows. The new method implements a space-time spectrally accurate method in the bulk regions of a multi-phase/multi-material flow and implements the cell integrated semi-Lagrangian moment-of-fluid method in the vicinity of mixed material computational cells. In the new method, the space-time order can be prescribed to be 2 ≤ p (x) ≤ 16 (space) and 2 ≤ p (t) ≤ 16 (time) respectively. represents the adaptive mesh refinement level. Regardless of the space-time order, only one ghost layer of cells is communicated between neighboring grid patches that are on different compute nodes or different adaptive levels . The new method is first tested on incompressible vortical flow benchmark tests, then the new method is tested on the following incompressible multi-phase/multi-material problems: (i) vortex shedding past a tilted cone and (ii) atomization and spray of a liquid jet in a gas cross-flow.
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