Round Table Discussions (I): Issues in Hydrodynamic Simulations

Michel has asked me to talk about fundamental issues in hydrodynamics. We heard many interesting applications but I would like to just stick to the issues of algorithms and fundamentals. I’ll just talk about these issues on the first slide and I’ll just leave this on and let you ask questions around these issues. The real issue, by the way, in hydrodynamics (I believe certainly for the people who try to develop lattice gas models) is to get into the high Reynolds number turbulent regime. As we have heard, that is still an issue which, I think, is the central unsolved problem in classical hydrodynamics. The traditional approach of course, is via the Navier-Stokes equations that are discretized. I mean that’s the traditional way of doing numerical hydrodynamics with discretized models: finite difference, finite elements, spectral methods. Let me just point out that these numerical schemes at the moment are more or less still confined to 2 dimensional systems. The computers, only by the end of this decade, will be probably able to achieve significant 3 dimensional calculations. The other approach we have heard about is Boltzmann. Let me point out that from linearized Boltzmann, one can go rigorously to Navier-Stokes. The solution of the linearized Boltzmann equation is equivalent to the Navier-Stokes equations but it deals of course with the additional variable of the velocity. There are several methods by which one can solve the Bolzmann equation. We have not heard them all yet but we will hear more tomorrow. I talked briefly about the discretized lattice Bolzmann equation, where you discretize the right hand side of the equation, into a lattice both spatially resolved and velocity resolved on a rather crude level. There is also the particle method solution of the Boltzmann equation which Bird talked about-probably people did not follow too well what the principles of that method were but we can go back to that.