Direct Numerical Simulations of Flows With Phase Change

During the last decade, direct numerical simulations of multiphase flow have emerged as a major research tool. It is now possible, for example, to simulate the motion of several hundred bubbles and particles in simple flows and to obtain meaningful average quantities that can be compared with experimental results. These systems are, however, still very simple compared to those systems routinely encountered in engineering applications. It is, in particular, frequently necessary to account for phase change, both between solid and liquid as well as liquid and vapor. Most materials used for manmade artifacts are processed as liquids at some stage, for example, and the way solidification takes place generally has major impact on the properties of the final product. The formation of microstructures, where some parts of the melt solidify faster than others, or solidify with different composition as in the case of binary alloys, is particularly important since the size and composition of the microstructure impact the hardness and ductility, for example, of the final product. Boiling is one of the most efficient ways of removing heat from a solid surface. It is therefore commonly used in energy generation and refrigeration. The large volume change and the high temperatures involved can make the consequences of design or operational errors catastrophic and accurate predictions are highly desirable. The change of phase from liquid to vapor and vice-versa usually takes place in a highly unsteady manner with a very convoluted phase boundary. Numerical simulations are therefore essential for theoretical investigations and while a few simulations of both problems have been published, the field is still very immature. In the talk the author gives a brief overview of the state of the art and discusses recent simulations of boiling and solidification in some detail. The progress made during the last few years in simulating the motion of multiphase flows without phase change has relied heavily on the so-called “one-fluid” formulation of the governing equations. In this approach one set of equations is written for all the phases involved. The formulation allows for different material properties in each phase and singular terms must be added at the phase boundaries to correctly incorporate the appropriate boundary conditions. The key challenge is to correctly advect the phase boundary and a number of methods have been proposed to do so. Those include the Volume-Of-Fluid (VOF), the level-set, the phase field methods, as well as front-tracking methods where the boundary is explicitly tracked by connected marker points [1]. The last approach, front tracking, has been particularly successful and is used for the examples shown here. In both boiling and solidification it is necessary to solve the energy equation, in addition to conservation equations for mass and momentum, and account for the release/absorption of latent heat at the phase boundary. The latent heat source also determines the motion of the phase boundary relative to the fluid. In boiling there is significant volume expansion as liquid is transformed into vapor and this expansion must be accounted for in the mass conservation equation. For solidification the volume expansion can often be neglected, but the transformation of the liquid into a stationary solid poses new computational challenges. An example of a bubble undergoing vapor explosion is shown in figure 1. The bubble is initially started as a small nearly spherical sphere in superheated liquid confined in a domain that is periodic in two directions, with a solid wall at the bottom and open on the top to allow outflow as the bubble expands. In this case the domain is resolved by a 643 grid. As the bubble grows, the interface becomes unstable, developing a corrugated shape (usually referred to experimentally as a “black bubble” since the corrugated surface is opaque). The increase in surface area greatly affects the growth rate of the bubble. Figure 2 shows one example of a simulation of the growth of a dendrite of pure material in uniform flow. The domain is a square resolved by a 2563 grid. A uniform inflow is specified on the left boundary, the top and bottom boundaries are periodic, and all gradients are set to zero at the outlet boundary. The temperature of the incoming flow is equal to the undercooled temperature and as latent heat is released at the phase boundary, the flow sweeps it from the front to the back. This results in a thinner thermal boundary layer at the tip of the upstream growing arm and a relatively uniform temperature in the wake. The growth rate of the upstream arm is therefore enhanced and the growth of the downstream arm is reduced.