2D-3V Visualization of Non-Equilibrium Flows

Visualization plays important role in the analysis and interpretation of numerical solutions. The development of flow visualization techniques and tools for rendering scalar, vector and tensor fields obtained by Computational Fluid Dynamics (CFD) techniques has significantly accelerated the adaptation of CFD in many areas of science and engineering. Visualizing higher-dimensional (> 3D) data has been an active research area in order to design methods that overcome the difficulties involved in representing such data on a two-dimensional computer screen. Higher-dimensional data occur in a wide range of applications including medical imaging, uncertainty visualization and fluid flows. Non-equilibrium flows encountered in supersonic flight at high altitudes, flows expanding into vacuum and flows in microdevices are governed by the Boltzmann equation for the the velocity distribution function(VDF) that, in general, depends on 7 independent variables - time, 3 physical coordinates, and 3 velocity coordinates. Unlike continuum equations, e.g. the Navier-Stokes equations, for the macroscopic parameters such as density, velocity, temperature and pressure, the Boltzmann equation is in terms of the velocity distribution function. In spite of the VDF being the fundamental quantity of interest, visualization techniques applied to non-equilibrium flow problems solved using the Boltzmann equation and its approximations have been restricted to macroscopic parameters mainly due to the high dimensional nature of the VDF. The macroscopic parameters, which are moments of the VDF, do not completely describe the features of non-equilibrium flows. The main goal of this work is to study the distinguishing features of non-equilibrium flows by probing the VDF using visualization techniques for 3-D scalar fields.

Figure 1 : Density contours for hypersonic flow of nitrogen past a flat plate for M = 10

Figure 1 shows the density contours for hypersonic flow of nitrogen over a flat plate. The freestream Mach number is 10 and the shock structure can clearly be seen. The first order approximation to the velocity distribution function at various locations labeled 1, 2, and 3 in figure 1 can be obtained using the Chapman-Enskog theory by computing the shear stress and heat flux using the macroscopic quantities. The distribution functions hence obtained are shown below in figure 2. The distribution function significantly deviates from the spherical (isotropic) Maxwellian distribution at the leading edge (location 1) where the rarefaction effects are expected to be the maximum. As we move to location 2, the degree of non-equilibrium decreases and location 3 is very close to equilibrium. The distribution functions within a shock wave obtained using various methods including Chapman-Enskog theory, direct simulation Monte Carlo (DSMC), deterministic ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK), and Mott-Smith approximate theory were compared with each other[1].

Figure 2 : Isosurfaces of velocity distribution functions at location 1 (left), location 2 (middle), and location 3 (right)

References :

[1] A. Venkattraman and A.A. Alexeenko, "Visualizing Non-Equilibrium Flow Simulations using 3-D Velocity Distribution Functions", Proceedings of the 27th International Symposium on Rarefied Gas Dynamics, Pacific Grove, California, July 2010.