The gas dynamics under gravitational field is usually associated with the multiple scale nature due to large density variation and a wide range of local Knudsen number. It is challenging to construct a reliable numerical algorithm to accurately capture the non-equilibrium physical effect in different regimes. In this paper, a well-balanced unified gas-kinetic scheme (UGKS) for all flow regimes under gravitational field will be developed, which can be used for the study of non-equilibrium gravitational gas system. The well-balanced scheme here is defined as a method to evolve an isolated gravitational system under any initial condition to an isothermal hydrostatic equilibrium state and to keep such a solution. To preserve such a property is important for a numerical scheme, which can be used for the study of slowly evolving gravitational system, such as the formation of star and galaxy. Based on the Boltzmann model with external forcing term, an analytic time evolving (or scale-dependent) solution is constructed to provide the corresponding dynamics in the cell size and time step scale, which is subsequently used in the construction of UGKS. As a result, with the variation of the ratio between the numerical time step and local particle collision time, the UGKS is able to recover flow physics in different regimes and provides a continuum spectrum of gas dynamics. For the first time, the flow physics of a gravitational system in the transition regime can be studied using the UGKS, and the non-equilibrium phenomena in such a gravitational system can be clearly identified. Many numerical examples will be used to validate the scheme. New physical observation, such as the correlation between the gravitational field and the heat flux in the transition regime, will be presented. The current method provides an indispensable tool for the study of non-equilibrium gravitational system.

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The gas dynamics under external force field is essentially associated with multiple scale nature due to the large variations of density and local Knudsen number. Single scale fluid dynamic equations, such as the Boltzmann and Navier-Stokes equations, are valid in their respective modeling scales, and it is challenging for the modeling and computation of a multiple scale problem across different regimes and capture the corresponding non-equilibrium flow physics. Based on the direct modeling of conservation laws in the discretized space, a well-balanced unified gas-kinetic scheme (UGKS) for multiscale flow transport under external force field has been developed and is used in the current study of non-equilibrium gaseous flow under external force field. With the variation of modeling scale, i.e., the cell size and time step, the UGKS is able to recover cross-scale flow physics from particle transport to hydrodynamic wave propagation. Theoretical analysis based on the kinetic model equation is presented to conceptually illustrate the effects of external force on the non-equilibrium heat transport. The heat conduction problem in the near-equilibrium regime due to the external forcing term is quantitatively investigated. In the lid-driven cavity flow study, the stratified flow is observed under external force field. With the increment of external force, the flow topological structure changes dramatically, and the temperature gradient, shearing stress, and external force play different roles in the determination of the heat flux in different layers corresponding to different flow regimes. As a typical non-Fourier's heat effect in the transition regime, the additional external force enhances the heat flux significantly along the forcing direction, and the relationship ∆q ∝ ∇Φ, where q is the heat flux and Φ is the external force potential, is fully confirmed in the flow regimes with non-vanishing effect of particle mean free path. Through the numerical experiment, it is clear that the external force plays an important role in the dynamic process of non-equilibrium flow transport and heat transfer. *