Session E19: Rarefied Gases and DSMC

Study of shock-shock interactions for a Double Wedge using the DSMC Approach

The Direct Simulation Monte Carlo (DSMC) method, an approach for modeling finite-Knudsen number flows is being used to study the laminar, shock-shock interactions from hypersonic flows about a double-wedge configuration in the Hypervelocity Expansion Tube (HET) facility. The study focuses on the investigation of Mach 7 nitrogen flows about a 30-/55-deg double wedge model for stagnation enthalpy of 8.0 MJ/kg. Schlierens are generated to visualize the shock structure and shock-shock interactions present in these flows and are compared with the experimental images. The computed heat transfer values from the simulations match the experiment along the first surface, but on the second wedge the computed heat transfer distribution over predicts the measured peak values. The influence of different models for nonequilibrium nitrogen dissociation, rotational and vibrational relaxation rates, and gas-surface interactions on the shock interaction region are analyzed for high enthalpy flow features and heat transfer rates. Overall good agreement is observed in the experimental and computational results. Studies are being performed related to flow unsteadiness and three-dimensional affects to resolve remaining discrepancies between measurements and modeling.   

Measurement of Tangential Momentum Accommodation Coefficient (TMAC) using a Disc Spin-Down Experiment in Low Pressure Gas

The objective is the measurement of TMAC for gas versus surface interactions. An experimental facility is built to accommodate a disc spin-down experiment in various gas pressures. The experiment measures the drag on the surface of the disc through measurement of its rotational speed during spin-down. Computational fluid dynamics is used to determine an acceptable shape and size of the facility and to estimate the pressure at which free molecular flow regime may be reached. The spin-down speed is translated to angular deceleration. Torque is obtained from disc moment of inertia and the angular deceleration. Data shows that the torque is a linear function of angular velocity. Torque is non-dimensionalized and is plotted against Reynolds number (Re). Between atmospheric pressure and a pressure of 357 Pa, the non-dimensional torque decreases with Re. At 2.7 Pa, the non-dimensional torque does not show any change with Re and the system presumably attains continuum breakdown. At a pressure of 0.71 Pa the free molecular flow regime is reached. The measured TMAC between air and aluminum shows the range between 0.7209 and 0.7355. Future work will measure the TMAC of materials commonly used in aerospace systems such as titanium, kapton, and carbon fiber.   

Maximum-entropy reconstruction method for moment-based solution of the Boltzmann equation

We describe a method for a moment-based solution of the Boltzmann equation. This starts with moment equations for a $10+9\,N, N=0,1,2...$-moment representation. The partial-differential equations (PDEs) for these moments are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy construction of the velocity distribution function $f({\bf c},{\bf x},t)$, using the known moments, within a finite-box domain of single-particle-velocity (${\bf c}$) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, {\it Continuum Mech. Thermodyn.}, 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using a Monte-Carlo method. This allows integration of the moment PDEs in time. Illustrative examples will include zero-space- dimensional relaxation of $f({\bf c},t)$ from a Mott-Smith-like initial condition toward equilibrium and one-space dimensional, finite Knudsen number, planar Couette flow. Comparison with results using the direct-simulation Monte-Carlo method will be presented.   

Shock Structure at Moderate and Large Mach Numbers

The structure of gas-dynamic shock waves at hypersonic conditions is of great interest. The Navier-Stokes formulation is known to yield incorrect shock profiles even at moderate Mach numbers. This is an excellent test problem for extensions of such equations since excellent experimental results are available. Continuum theories, and indeed most statistical mechanics theories, that have been advanced to reconcile such discrepancies have not been fully successful. Here, we present a second order formulation of the governing stress tensor and heat flux based solely on a continuum formulation. The constitutive equations for the gas, in addition to the known transport properties, also introduce additional viscosity and thermal conductivities which generally depend on density and temperature. Their specific dependence are estimated from kinetic theory. Results of the second-order equations applied to the shock structure are obtained for monatomic and diatomic gases over a large range of Mach numbers and are compared to experimental results.   

Thermal transpiration of a rarefied gas between parallel plates with different accommodation coefficients

Thermal transpiration of a rarefied gas between parallel plates with different accommodation coefficients is studied on the basis of the linearized Boltzmann equation for a hard sphere molecular gas. The Boltzmann equation is solved numerically using a finite difference method. The macroscopic variables as well as the mass flow rate of the gas are studied over a wide range of the Knudsen number (the mean free path divided by the channel width) and the two accommodation coefficients. When the Knudsen number is not so small, the mass flow rate of the gas increases as the accommodation coefficients decrease. When the Knudsen number is sufficiently small, however, the tendency is opposite and the mass flow rate basically decreases as the accommodation coefficients decrease. The solution for infinitesimally small accommodation coefficients is also discussed.   

The role of divergences for shock waves

Several continuum theories for shock waves give rise to a set of differential equations in which the analysis of the underlying vector field can be done using the tools of the theory of dynamical systems. We illustrate the importance of the divergences associated with the vector field by considering the ideas by Maxwell and Cattaneo and applied them to study shock waves in dilute gases. Different theoretical descriptions for shock waves are mentioned and some of them are compared with experimental data and computer simulations. Our goal is to derive conditions under which the shock wave problem has a solution by analyzing the singularities of the vector field.