An Artificial Viscosity for the III-Posed One-Dimensional Incompressible Two-Fluid Model

Abstract

Methods to remedy the ill-posedness of the basic one-dimensional two-fluid model, which is widely used in nuclear reactor safety codes, have been the subject of considerable study. Both of the two prevalent methods have drawbacks. Unconditional hyperbolization uses nonphysical constitutive relations to create a well-posed two-fluid model that is hyperbolic over all flow conditions. However, when the model is hyperbolized, it is also stabilized, which is not a universal property of two-phase flows. The second method, the preferred method of the U.S. Nuclear Regulatory Commission safety codes, is to simply use a first-order upwind numerical method that relies on numerical viscosity to regularize the ill-posedness of the model by damping the short-wavelength instabilities. Unfortunately, the scale of the “short wavelength” is related to a particular numerical grid or discretization. Because of the consistency of the numerical method, in the limit of an infinitely resolved grid, i.e., Δx, Δt→0, the numerical viscosity vanishes, as does its regularization effect. This results in a somewhat heuristic user guideline that suggests a lower limit on the grid size based on a cross-sectional dimension that is a combination of the long-wavelength assumption and experience. However, a cutoff wavelength achieved by numerical viscosity is not set by the grid size alone but also depends on the time step, the material, and the flow properties, as demonstrated with a von Neumann stability analysis. This can create poor resolution in areas where numerical stability may not be a substantial problem, unless the guideline is intentionally violated. Additionally, strict observance of this limit makes verification by convergence difficult or impossible. Therefore, it is proposed that an artificial viscosity be prescribed explicitly, i.e., independently of any particular numerical method or grid. An artificial viscosity model is derived that prescribes exactly a cutoff in the linear stability growth rate at a specified wavelength, e.g., λ0 = 2DH consistent with the aforementioned user guideline. It is shown, using the water faucet problem, that the proposed artificial viscosity model can be used to remove the high-frequency component of the solution without limiting the resolution of the grid. Furthermore, the solution also converges, which was not the case without the artificial viscosity.