精细平衡气体动理学格式

流体系统在重力作用下，流体压力可与重力相平衡，达到静力平衡状态.常规的有限差分和有限体积方法无法在离散尺度上保持这种平衡态，常常产生虚假速度等非物理现象.通过将重力源项的体积分改写为网格界面值的加权平均，实现离散条件下压力与重力的精细平衡，得到一种Navier-Stokes （NS）方程的精细平衡气体动理学格式.该格式可以在机器精度上精确地保持等温条件下的静力平衡态以及捕捉平衡态附近的小扰动传播.同时，该格式还能求解自然界中更常见的非等温平衡态.此类平衡态除了要求静力平衡还要求热流平衡.利用提出的格式求解NS方程，可以得到静止（机器精度下的零速度）的非等温平衡态，并且密度和温度分布都具有二阶精度.通过多个算例验证格式的有效性，表明该格式可更好地模拟重力场下的温度、密度的分层流动.

A Well-balanced Gas Kinetic Scheme

Hydrostatic equilibrium is a static state that the fluid pressure is balanced with gravity. Classic finite difference and finite volume methods cannot maintain this equilibrium state on discrete scales, and produce non-physical phenomena often such as spurious currents. In this work, volume integral of the source term is implemented at cell interface, so that the balance of pressure and gravity under discrete conditions is achieved on discrete scales. A well-balanced gas kinetic scheme of the Navier-Stokes(NS) equation is therefore constructed. The scheme maintains accurately the static equilibrium state under isothermal conditions and captures small perturbation propagation near equilibrium state at machine precision. At the same time, the scheme solves non-isothermal stratified flow near equilibrium state. Apart from the non-isothermal equilibrium described by Euler's equation, the scheme can also represent balance of thermal conduction in addition to force balance. It is shown that the scheme preserves second-order accuracy with respect to both density and temperature distribution when simulating an non-isothermal equilibrium state. Numerical examples reveal that the scheme is a potential tool in simulating stratified flow of temperature and density under gravitational field.