This proposal concerns the development and implementation of algorithms for compressible multimaterial and multiphase flows. In both types of flows, a major numerical stumbling block is accounting correctly for thermodynamical relaxation processes and equilibrium between the fluid components. Failure to do so is the reason why state-of-the-art single fluid algorithms often do not work in multifluid flows. Karni was among the first to realize that the key to robust multifluid algorithms is good control over the pressure field, and has been active in developing and implementing algorithms for complex multimaterial flows. It is proposed: (i) to develop multimaterial algorithms for stiff fluids with surface tension. Numerical issues such as stiffness, singular source treatment, interface sharpening and efficient time integration will be studied. Comparison with asymptotic theories and experiments of oscillating bubbles will be conducted; (ii) to extend multimaterial algorithms to multiphase flow models, in particular to the 2-velocity 2-pressure multiphase model which is sufficiently general and is time hyperbolic. Numerical aspects such as global conservation, positivity, efficient integration of stiff source terms will be addressed. The methods will be validated against numerical benchmark tests. (iii) to use multimaterial computations to 'validate' multiphase flow models by numerical averaging. Studying mean flow properties might provide numerical Hugoniot curves for the multiphase (nonconservative) systems, and might shed light on the controversial issue of constitutive closures. This part will be done in collaboration with H.M. Glaz of the University of Maryland.
Multimaterial and multiphase flows are flows consisting of several 'pure' gas or liquid components. For example, hydrogen and air, air in water, liquid droplets in air or even dust in air. Depending on the application, the interest may be to follow the motion of the interface that separates the different fluids and study its dynamics and stability. For example, computing the dynamics of a an oscillating gas bubble in water is useful for understanding the propagation of sound waves underwater, and for studying underwater explosions. In order to burn fuel efficiently in combustion chambers one needs to understand mixing processes, which in turn are governed by the dynamics of, say, a hydrogen jet in air. In some other applications, the number of interfaces may be huge and following the dynamics of individual interfaces is not only impossible but is also not interesting. For example, in bubbly liquids such as soda cans, in liquid suspensions such as in sprays or in dusty gases, the motion of a single bubble or droplet is not of interest. Rather, the interest is in understanding how the fluid mixture behaves 'on average'. Both types of flows lead to mathematical models that need to be solved on computers, and are computationally very intensive. They are also not easy to compute. The 'average' models are also mathematically less well understood. Experiments, while available, are difficult to conduct and are restricted by the accuracy of the instruments and their often large margins of error. Computer simulations provide therefore a complementary tool which may shed light and give insight into the complex problems of interest. The past few years have seen a growing interest in developing methods suitable for computing multimaterial/multiphase flows and in their efficient implementation to studying complex flow phenomena. This project concerns further development and implementation of such methods.