# Comparison of Navier-Stokes-Fourier Equation and Grad's Moment Equation Solutions for Turbulence

The accurate simulation and modeling of turbulent fluid motion is a very important research subject in natural and engineering sciences. Ranging from the application in aircraft design, the aeroacoustic optimization of an automobile and the cooling of modern microprocessor devises, the efficient simulation of turbulence is a key technology in modern engineering and design of technical products and devices.

The Boltzmann equation for the density distribution of particles in a fluid is the most fundamental equation of non-equilibrium gas kinetics and thus fluid motion. Due to its generality however, the numerical solution of the Boltzmann equation is only feasible for systems with low total particle count, e.g. rarefied gases.

For cases with higher particle counts, the equilibrium assumption is often valid, which allows the use of the well known equations of gasdynamics, the Navier-Stokes-Fourier system, which is the current state of the art model for turbulence fluid motion. In between these two regimes and overlapping them lies a physical regime where the validity of the equilibrium assumption is not clear-cut. In here, the numerical solution of the Boltzmann equation is still too computationally demanding, while the Navier-Stokes-Fourier system loses its validity. One way of tackling this problem is finding better mathematical descriptions of this regime, i.e. different sets of equations, often termed the "extended gasdynamics models''.

The goal of this work is to take a first significant step towards the evaluation of those extended gasdynamic models, such as e.g. the Grad 13 and the regularized Grad 13 equations, for the simulation and modeling of turbulent fluid motion.

The difficulty lies in the nature of turbulence: it is a highly chaotic phenomenon, which features a broad range of spatial and temporal scales, see Figure 1 for an example.

The approach in this project is to use direct numerical simulation of those model equations as a virtual experimental environment. To factor out numerical artifacts in the assessment of the results, a high resolution method was developed with a special emphasis on high performance computing. The multiscale nature of turbulence demands the largest available computing resources and simulations with O(10^{5}) processors were performed to assess the difference on the JSC Blue-Gene system JUQUEEN.

Source: © Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart

**Project Team Members:**

Gregor Gassner *(Mathematical Institute, University of Cologne, D-50931 Cologne, Germany)*

Manuel Torrilhon *(Center for Computational Engineering Science, RWTH Aachen, D-52062 Aachen, Germany)*

Andrea Beck, Sophie Knechtel, Thomas Bolemann *(Institute for Aerodynamics and Gasdynamics, Universität Stuttgart, 70569 Stuttgart, Germany)*

Prof. Dr.-Ing. Gregor Gassner

Mathematisches Institut, Universität zu Köln

Weyertal 86-90, D-50931 Köln/Germany

e-mail: ggassner@math.uni-koeln.de