Modeling of Viscous Shock Tube Using ES-BGK Model Kinetic Equations

Abstract

The viscous effects on unsteady shock wave propagation are investigated by numerical solution of the Boltzmann model kinetic equations. The kinetic equations are solved for two unsteady non-equilibrium flow problems, namely, the one-dimensional Riemann problem and a two-dimensional viscous shock-tube. The numerical method comprises the discrete velocity method in the velocity space and the finite volume discretization in physical space using various flux schemes. The discrete version of H-theorem is applied for analysis of accuracy of the numerical solution as well as of the onset of non-equilibrium. Simulations show that the maximum entropy generation rate in viscous shock tube occurs in the boundary layer / shock wave interaction region. The entropy generation rate is used to determine the time-variation of the speed of propagation of shock, contact discontinuity and rarefaction waves.

Entropy considerations in numerical simulations of non-equilibrium rarefied flows

“…Thus, entropy generation rate can be a useful quantity in calculating the location and velocity of both the shock wave and contact discontinuity. This can be extended further to study the rarefaction effects and shock speed attenuation by boundary layer interaction in 2D shock tube [25] as discussed below.…”

A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations

“…The ESBGK equilibrium distribution function is of the form [34]: Eqs. (26) and (27), the ESBGK model has ten unknowns α 1 -α 10 . In addition to solving for the conservation of mass and momentum, we solve for the conservation of six additional higher moments of equilibrium distribution function to obtain these coefficients [34]:…”

“…(13), (14) and Eqs. (26), (27). The central idea behind COMET is to couple the velocity ordinates strongly through a point-implicit direct solution of the distribution function values in each cell, and to traverse the spatial cells in a Gauss-Seidel procedure.…”

“…Subsequently, Cercignani and Tironi [26] studied Fourier flows to investigate the accuracy of ESBGK models. Over the last few years, ESBGK models have been used in a wide range of problems such as low pressure shock-tube [27], spacecraft re-entry [28], and squeeze film damping in microstructures [29]. The ESBGK model has also been found to accurately predict heat and tangential momentum transport between parallel walls [30].…”

Unified theory of gas damping of flexible microcantilevers at low ambient pressures

“…The discrepancy at higher pressures ͑p Ͼ 2700 Pa͒ occurs because the current model neglects the unsteady gas inertia term. An unsteady Boltzmann equation model 8,16 or a continuum regime slip-flow model 5 is expected to work better for such high pressures.…”