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Gas Dynamics

For the most technical processes, the motion of fluids is well described by continuum models, such as the Navier-Stokes equations. This is the case because under those common macroscopic conditions the fluid particles interact much more often among each other than with the surfaces, interfaces and boundaries of the fluid. If this situation changes, e.g. because the mean free path of the particles becomes very large or because characteristic flow lengths become very small, classical continuum models make wrong predictions of the flow fields. These rarefied flows are well described by the Boltzmann equation. Numerical methods to solve this integro-differential equation are well known for some time now.

In the transition regime, however, where we have a flow situation where a description by the Navier-Stokes equations does not suffice but a description by the full Boltzmann equation does not seem adequate, there is a gap in our ability to make theoretical predictions of flow conditions based on numerical simulations. This is because the well-understood continuum models do not hold any longer, but solution of the Boltzmann equation requires too expensive numerical effort. The rarefied regime, however plays an important role e.g. in the development of Microscopic Electro-Mechanical Systems where flow conditions can change between the continuum regime and the rarefied regime, like for read/write heads of hard disk drives.

Our research aims at theoretical descriptions and numerical methods that could be used to implement methods that not only allow for adaptation of the numerical solver to the flow fields, but eventually also to adapt the details of the physical model to the particular flow regime.

Recent Publications

Power law rank-abundance models for marine phage communities
Hoffmann, K. H. and Rodriguez-Brito, B. and Breitbart, M. and Bangor, D. and Angly, F. and Felts, B. and Nulton, J. and Rohwer, F. and Salamon, P.
FEMS Mircobiology Letters 273: 224-228 (2007) ; DOI: 10.1111/j.1574-6968.2007.00790.x

Metagenomic analyses suggest that the rank-abundance curve for marine phage communities follows a power law distribution. A new type of power law dependence based on a simple model in which a modified version of Lotka-Volterra predator-prey dynamics is sampled uniformly in time is presented. Biologi- cally, the model embodies a kill the winner hypothesis and a neutral evolution hypothesis. The model can match observed power law distributions and uses very few parameters that are readily identifiable and characterize phage ecosystems. The model makes new untested predictions: (1) it is unlikely that the most abundant phage genotype will be the same at different time points and (2) the long-term decay of isolated phage populations follows a power law.

The cumulant method for gas dynamics
Seeger, S. and Hoffmann, K. H. and Meyer, A.
Parallel algorithms and cluster computing : 335 Springer-Verlag, Berlin Heidelberg, 2006 ; ISBN: 3-540-33539-0

On symbolic derivation of the cumulant equations
Seeger, S. and Hoffmann, K. H.
Computer Physics Communications 168(3): 165--176 (2005) ; ISSN: 0010-4655, DOI: 10.1016/j.cpc.2005.03.106

We discuss the application of Mathematica for automated, symbolic calculation of the cumulant equations of arbitrary order. Like moment equations, these partial differential equations-describing fluid motion on a mesoscopic scale-may be considered an approximation to the Boltzmann equation, a highly nonlinear integro-differential equation that describes the motion of gases at a microscopic scale. Though the cumulant method provides a simple and compact presentation of the theory, actual calculation of very high order equations turns out to be a challenging task.

On the Domain of Hyperbolicity of the Cumulant Equations
Seeger, S. and Hoffmann, K. H.
Journal of Statistical Physics 121(1--2): 75--90 (2005) ; ISSN: 0022-4715; DOI: 10.1007/s10955-005-6969-2

In this article we consider the influence of non-equilibirum values of classical variables on the eigenvalues of the advection part of the cumulant equations. Real and finite eigenvalues are a neccessary condition for the cumulant equations to be hyperbolic which can be used to obtain estimates on admissible deviations from equilibrium for a model of particular order still to be valid. We find that this condition puts no constraints on velocity and shear stress values, but specific energy must be positive, normal stress must be bounded by specific energy and heat flux not be too large.

The cumulant method for the space-homogeneous Boltzmann equation
Seeger, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 17(1): 51--60 (2005) ; ISSN: 0935-1175, DOI: 10.1007/s00161-004-0187-z

In this work we give a comparison of the exact Bobylev/Krook-Wu solution to the space-homogeneous Boltzmann equation and numerical results obtained by a implementation of the cumulant method for the space-homogeneous case. We find excellent agreement of the numerical solution to the cumulant equations with the exact solution of the space-homogeneous Boltzmann equation as long as the exact, non-linear production terms are used. If a linearized variant of the production terms is used, relaxation rates may be underestimated due to convergence to the solution of the linearized equations.

Erratum: The cumulant method applied to a mixture of Maxwell gases
Seeger, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 16(5): 515 (2004) ; DOI: 10.1007/s00161-004-0183-3

Anwendung der Methode der gewichteten Residuen auf die eindimensionale Boltzmanngleichung
Balg, Janett
Bachelorarbeit an der TU-Chemnitz, 2004

The Cumulant Method
Seeger, S.
PhD Thesis, Chemnitz University of Technology, 2003

In this work, a new method to reduce Boltzmann equation to a system of partial differential equations is discussed. After a short introduction to kinetic theory of an inert mixture of gases an overview of the various moment methods known from literature is given. The cumulant method, presented in the following, is based on the assumption that due to the collision processes in a gas correlations of higher order decay more rapidly than correlations of lower order. Based on this assumption the eautions of motion for the cumulants are derived and the production terms of the resulting balance equations are calculated for a mixture of inter Maxwell gases are calculated. Examination of the relexation to an equilibrium state allows to relate these equations to models known from continuum mechanics and shows the validity of the assumption made for this case. In the second part results of numerical experiments are presented, where simulations with various boundary conditions are carried out for Couette and Poiseulle flows. Depending on the particular boundary conditions applied, both characteristic properties of rearefied gases but also of Navier-Stokes flows are observed. The last part of this work discusses moment equations as a particular form of the method of weighted residuals applied to the Boltzmann equation, giving an outlook to future work.

The cumulant method applied to a mixture of Maxwell gases
Seeger, S. and Hoffmann, K. H.
Continuum Mechanics and Thermodynamics 14(2): 321--335 (2002) ; ISSN: 0935-1175

We apply the recently proposed cumulant method to derive the production terms for a mixture of gases of Maxwell-molecules in two and three dimensions. For the single component Maxwell gas we introduce a linear approximation of the production terms and give an analytical solution for the (space-)homogeneous case. We find that the eigenvariables of the linearized productions appear in three different kinds and the first few can be related to classical thermodynamic quantities.

Computational Statistical Physics
Hoffmann, K. H. and Schreiber, M.
Springer Verlag, Berlin, 2002 ; ISBN: 3-540-42160-2

In recent years statistical physics has made significant progress as a result of advances in numerical techniques. While good textbooks exist on the general aspects of statistical physics, the numerical methods and the new developments based on large-scale computing are not usually adequately presented. In this book 16 experts describe the application of methods of statistical physics to various areas in physics such as disordered materials, quasicrystals, semiconductors, and also to other areas beyond physics, such as financial markets, game theory, evolution, and traffic planning, in which statistical physics has recently become significant. In this way the universality of the underlying concepts and methods such as fractals, random matrix theory, time series, neural networks, evolutionary algorithms, becomes clear. The topics are covered by introductory, tutorial presentations.