Simulation of shock wave diffraction by a square cylinder in gases of arbitrary statistics using a semiclassical Boltzmann–Bhatnagar–Gross–Krook equation solver

Abstract

The unsteady shock wave diffraction by a square cylinder in gases of arbitrary particle statistics is simulated using an accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space. The numerical method is based on the discrete ordinate method for discretizing the velocity space of the distribution function and high-resolution method is used for evolving the solution in physical space and time. The specular reflection surface boundary condition is employed. The complete diffraction patterns including regular reflection, triple Mach reflection, slip lines, vortices and their complex nonlinear manifestations are recorded using various flow property contours. Different ranges of relaxation times corresponding to different flow regimes are considered, and the equilibrium Euler limit solution is also computed for comparison. The effects of gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics are examined and depicted.

BGK model of the multi-species Uehling-Uhlenbeck equation

“…But due to the complexity of the collision operator, which is a serious obstacle to practical application of the equation, and relaxation time approximations, or quantum BGK models are widely used to understand the transport phenomena and compute transport coefficients for semi-conductor device and crystal lattice [2,20,33,34,35,36,44,50,51] and various flow problems involving quantum effects [15,22,23,33,45,55,56,58,63,64]. For the development of numerical methods for quantum BGK model, we refer to [15,22,23,46,52,56,59,63,64,65]. We mention that the prototype of relaxation type models in quantum theory can be traced back to the Drude model [18,19] which successfully explained the fundamental transport property of electrons such as the Ohm's law or Hall effect.…”

Stationary Quantum BGK Model for Bosons and Fermions in a Bounded Interval

“…The stationary quantum BGK model [27,36,37,39,40,42,48,49,51] in a bounded interval reads The momentum distribution function f (x, p) depends on the position x ∈ [0, 1] and the momentum p ∈ R 3 . The Knudsen number τ > 0 measures how rarefied the gas system is, and is defined by the ratio between the characteristic length and mean free path.…”

Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation

“…In the classical rarefied gas flow computation, the implementation of a discrete-ordinate method and high-resolution total variation diminishing (TVD) [29] and essentially non-oscillatory (ENO) schemes [30] to nonlinear model Boltzmann equations has been developed [31] and a similar work using a high-order compact scheme has been proposed [32]. Extension to semiclassical Boltzmann-BGK equation using TVD and weighted ENO (WENO) schemes [33] has been reported [34]. Such a direct method will allow one to examine the same physical flow problems but with different gas or particles.…”

“…The numerical flux for a second-order TVD scheme using flux limiter can be found in [28], and that for a fifth-order WENO scheme (WENO3, (r = 3)) can be found in [34] and will not be repeated here. The class of asymptotic preserving schemes for the kinetic equations and related problems with stiff sources can be beneficially applied as well [36,40].…”