  In the study of gas dynamics, theoretical modeling and numerical simulation
are mostly set up with deterministic settings. Given the coarse-grained
modeling in theories of fluids, considerable uncertainties may exist between
flow-field solutions and real-world physics. To study the emergence,
propagation and evolution of uncertainties from molecular to hydrodynamic level
poses great opportunities and challenges to develop both sound theories and
reliable multi-scale algorithms. In this paper, we study the stochastic
behavior of multi-scale gas dynamic systems, especially focusing on the
non-equilibrium effects. The theoretical analysis is presented on the basis of
kinetic model equation and its upscaling macroscopic system, with the
reformulation from the stochastic Galerkin method. A newly developed stochastic
kinetic scheme is employed to conduct numerical simulation of homogeneous
relaxation, normal shock structure, shear layer and lid-driven cavity problems.
Different kinds of uncertainties are involved in conjunction with the gas
evolutionary processes. New physical observations, such as the synergistic
propagation pattern between mean fields and uncertainties, sensitivity of
different orders of uncertainties, and the influence of boundary effects from
continuum to rarefied regimes, will be identified and analyzed theoretically.
The paper serves as a heuristic study of quantifying the uncertainties within
multi-scale flow dynamics.
