Minimal Kinetic Model for Simulating Hydrodynamics

The atomistic description of (classical) macroscopic systems is based on Newtonian mechanics. This theory is capable of describing phenomena occurring at nano-scale as well as those occurring at planetary scales. These equations are quite computer friendly, both in terms of implementation and performance. However, all direct application of Newtonian mechanics to the molecular scale inevitably leads to a gigantic wave of computational complexity due to the enormous number of atoms/molecules constituting macroscopic systems.

A kinetic description such as Boltzmann equation, in terms of one particle distribution function, is a useful bridge between a completely molecular description or a complete macroscopic description. However, a direct discretization of kinetic equations for solving hydrodynamic problems of engineering interests is still a monumental task. It is obvious that in order to have an efficient Boltzmann solver for hydrodynamic simulations important simplifications are needed at the level of modeling itself. Formulating coarse-grained continuum models circumvents this problem, where one writes partial differential equations describing the space-time evolution of a few macroscopic fields such as fluid density, pressure, temperature.Such a macroscopic descriptions are indeed quite successful in describing hydrodynamic problems of macroscopic length scales large compared with micrometers.

However, such a coarse representation shows clear limitations for two important applications:

• Hydrodynamics at micro/nano (sub-micrometer) scale even for gases
• Macroscopic systems out of equilibrium, e.g., fully turbulent flow

While the difficulties at micro/nano scale are due to failure of macro description, difficulties occurring for turbulence are about solving a highly non-linear system with large number of degrees of freedom.

The first problem describes a typical problem in multiscale modeling where a complete macroscopic description starts to breakdown but the domain is still too large for an efficient microscopic description.

In order to overcome the first difficulty usual modeling strategies are:

• ncorporation of more variables (extended thermodynamic formulation or Grad-13 moment representation);
• Writing more accurate closure schemes for macroscopic variable (Burnett like description);
• Hybrid micro-macro simulations.

The important practical problems faced by a description in term of extended set of macroscopic variables are lack of good boundary conditions and requirement of computationally extensive algorithms for a numerical simulation (mainly due to introduction of non-locality by higher order derivatives).

On the other hand a hybrid simulation schemes faces usual difficulties such as: what to do at micro-macro interface, when (or in which part of simulation domain) a macro description is sufficient and when it is important to switch to micro descriptions?Despite of recent progresses, hybrid simulations are yet to become practical simulation tool for microflow simulations. Further, the problem with macroscopic description in terms of Navier-Stokes equation of turbulence is related to the strong equation. While the enormous number of degree of freedoms in a fully developed turbulent flows of engineering interest, prohibits direct simulations of the phenomena, strong non-linearity of the equation ensures that a reduced description based on the scale-separation argument will not work.

Project

In this project, we had proposed an alternate strategy for overcoming problem faced by macroscopic descriptions based on the Navier-Stokes equation. Our recipe is to write model kinetic equations, sufficient to describe hydrodynamics at desired length scales, in the discrete form itself, which are consistent with the second law of thermodynamics and which retains the known computational efficiency of the Kinetic schemes. The proposed model are minimal kinetic model in the sense that even with a highly reduced set of variables, they mimic the hydrodynamics and thermodynamics inherent in the microscopic kinetic equation, while preserving the simplicity of a kinetic simulation algorithm.The non-linear stability and highly efficient performance coupled with ease of implementation and a fully parallel algorithmare few salient features of the new scheme.

Approach

The basic approach taken in the project is construction of discrete kinetic theories based on discrete H functions, relevant to the dynamics considered. The resulting models, by construction, demonstrate unconditional stability, a highly desirable property in any numerical scheme. The method is successfully employed for the simulation of microflows as well as turbulence. For turbulence simulations, the thermodynamics inherent in the discrete kinetic model act as built in sub-grid model.

Turbulence Simulations movie I | movie II

Related references »»

Contacts

Hans Christian Öttinger