Limitation principle for computational fluid dynamics

Abstract

Theoretical gas dynamics uses the physical Knudsen number $$\mathrm {Kn}_\mathrm{p}$$, which is defined as the ratio of the particle mean free path $$\lambda $$ to the characteristic length scale L, to categorize the flow into different regimes. The Boltzmann equation is the fundamental equation for dilute gases, while the Navier–Stokes (NS) equations are used for the description of continuum flow at $$\mathrm {Kn}_\mathrm{p}\le 10^{-3}$$. For computational fluid dynamics (CFD), the numerical resolution is limited by the discrete cell size and time step. Therefore, we can define a cell Knudsen number $$\mathrm {Kn}_\mathrm{c}$$ as the ratio of the particle mean free path $$\lambda $$ to the cell size $$\Delta x$$. In CFD, the numerical solution and the corresponding numerical flow regime are fully controlled by a numerical Knudsen number $$\mathrm {Kn}_\mathrm{n}$$, which is a function of the physical Knudsen number $$\mathrm {Kn}_\mathrm{p}$$ and the cell Knudsen number $$\mathrm {Kn}_\mathrm{c}$$. The limitation principle relates to the connections between $$\mathrm {Kn}_\mathrm{n}$$, $$\mathrm {Kn}_\mathrm{p}$$, and $$\mathrm {Kn}_\mathrm{c}$$. In this paper, based on the relationship between the modeling equation, cell resolution, and the physical structure thickness, we propose the division of numerical flow regimes. According to the limitation principle, the range of validity of the NS equations is extended to $$\max (\mathrm {Kn}_\mathrm{p},\mathrm {Kn}_\mathrm{c})\le 10^{-3}$$. During a mesh refinement process, in some cases the NS equations alone may not be able to capture the flow physics once the large gradients and high-frequency modes are resolved by numerical mesh size and time step. In order to obtain a physical solution in the corresponding numerical scale efficiently, a multiscale method is preferred to identify the flow physics in the corresponding cell Knudsen number $$\mathrm {Kn}_\mathrm{c}$$, such as capturing hydrodynamic wave propagation in the coarse mesh resolution case and the kinetic particle transport in the fine mesh case. The unified gas-kinetic scheme (UGKS) is such a multiscale method for providing continuum, near-continuum, and non-equilibrium solutions with a variation of cell Knudsen number. Numerical examples with different physical Knudsen numbers are calculated under different cell Knudsen numbers. These results show the mesh size effect on the numerical representation of a physical solution. In comparison with the NS and direct Boltzmann solvers, the multiscale UGKS is able to capture flow physics in different regimes seamlessly with a variation of numerical resolution.