In this work, we propose a flux solver that relies on a double distribution function (DDF) based discrete gas-kinetic scheme (DGKS) for incompressible and compressible viscous flows. By decomposing the phase space of the Maxwellian distribution function, the double distribution function (DDF) model is established to remove the phase energy variables and develop more compact formulations. It utilizes the density distribution function to recover the macroscopic mass and momentum conservations, and the energy distribution function to derive the macroscopic energy equation. Associated with a modified collision operator for the energy distribution function, the Prandtl number in the recovered energy equation becomes adjustable. The DDF model can be further simplified with the idea of the improved circular function and by replacing the continuous integration with the quadrature algorithm, which yields the discrete DDF model. This model can then be adopted in the reconstruction of numerical fluxes on the cell interface within the framework of the finite volume method, and the improved discrete gas-kinetic scheme (DGKS) is established. In contrast to the previous DGKS [39], the improved flux solver recovers the correct macroscopic equations and allows free adjustment of the Prandtl number. Besides, the introduction of the energy distribution function abbreviates the expression of the energy flux and thus makes the final formulations more compact. Numerical tests, including the compressible Couette flow, lid-driven cavity flow, transonic flow around the airfoil, shock-boundary layer interaction and supersonic flow around a ramp segment, are presented to validate the proposed flux solver in various flow conditions and compare its performance with the previous DGKS methods [39], [43].

在这项工作中，我们提出了一种通量求解器，它基于不可分配和可压缩粘性流的基于双分布函数（DDF）的离散气体动力学方案（DGKS）。通过分解麦克斯韦分布函数的相空间，建立了双分布函数（DDF）模型，以消除相能量变量并开发更紧凑的公式。利用密度分布函数恢复宏观质量和动量守恒，利用能量分布函数推导宏观能量方程。与用于能量分配函数的改进的碰撞算子相关联，恢复的能量方程式中的Prandtl数变为可调的。DDF模型可以通过改进循环函数的思想以及用正交算法代替连续积分来进一步简化，从而得到离散DDF模型。然后可以在有限体积方法的框架内，采用该模型重建单元界面上的数值通量，并建立改进的离散气体动力学方案（DGKS）。与之前的DGKS [39]相比，改进的磁通求解器恢复了正确的宏观方程，并允许自由调整Prandtl数。此外，能量分布函数的引入简化了能量通量的表达，从而使最终的配方更加紧凑。数值测试，包括可压缩的库埃特流，盖驱动的腔流，翼型周围的跨音速流，