Semiclassical Axisymmetric Lattice Boltzmann Method

A semiclassical lattice Boltzmann method is presented for axisymmetric flows of gas of particles of arbitrary statistics.The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in twodimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al.(Phys.Rev.E.,64(2001),011208)is adopted and forcing term is added into the resulting microdynamic evolution equation.The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form.The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion.Computations of uniform flow over a sphere to verify the method are included.The results also indicate distinct characteristics of the effects of quantum statistics.

Semiclassical Lattice Boltzmann Simulations of Rarefied Circular Pipe Flows

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method.The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al.[Phys.Rev.E.,64(2001),011208]is adopted by adding forcing terms into the resulting microdynamic evolution equation.The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form.The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion.The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented.It is found the Knudsen minimum can be captured in all three statistics studied.The results also indicate distinct characteristics of the effects of quantum statistics.

A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics

An accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics.The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term.The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time.The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann,Bose-Einstein and Fermi-Dirac statistics.Computational examples in one-and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved.Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method.The recovery of quantum statistics to the classical limit is also tested for small fugacity values.