High Resolution Finite Volume Method And A Refined V|OF Algorithm

Most TVD schemes available in the literature are reviewed and their design principles are also revealed. These schemes have been grouped into three br oad categories, viz. One-step Time-space-coupled Unsteady TVD(OTU-TVD), Multi-step Time-space-separated Unsteady TVD(MTU-TVD) and Semi-discrete Steady-state TVD(SS-TVD), based on the types of flux-limiters adopted. Furthermore, a refined SS-TVD limiter, referred to as TCDF, is proposed for steady-state calculations based on the review. Nine existing classical SS-TVD limiters are compared with the newly proposed TCDF in four two-dimensional pure-advection problems. The numerical results clearly show that TCDF results in a better overall performance in terms of accuracy and convergence.The NVD schemes, based on non-linear characteristics in the normalized variable diagram, are reviewed in detail. It is pointed out that the convection boundedness criterion(CBC) is employed for NVD schemes so as to provide boundedness. The explicit CBC proposed by Leonard and the implicit CBC by Gaskell and Lau are widely considered as the sufficient and necessary conditions to achieve the local boundedness for explicit and implicit NVD schemes respectively. It should be noted that all the NVD schemes considered in this paper can be converted into flux-limiter form to allow direct comparison, owing to the unique relationship between NVD and TVD spaces.When implementing TVD or NVD schemes, the nodal variable values at two upstream cells are required. For arbitrary unstructured grids, the far-upwind node is not readily available. Thus, it is not straightforward to extend TVD or NVD schemes into arbitrary unstructured grids. This study presents a detailed analysis of the existing far-upwind reconstruction techniques. Furthermore, a refined extrapolation reconstruction algorithm(New-Technique-2) and a refined interpolation reconstruction algorithm(FFISAM) are proposed and evaluated in order to more efficiently extend NVD and TVD schemes, respectively, into arbitrary unstructured meshes. The numerical results clearly show that the both New-Technique-2 and FFISAM lead to a better overall performance than the existing algorithms in terms of accuracy and convergence.With the aid of NVD schemes, a new algebraic-type VOF scheme(M-CICSAM), capable of capturing abrupt interfaces on arbitrary meshes, is proposed here. Without resort to any explicit interface reconstruction, M-CICSAM is able to precisely model the complex free surface deformation, such as interface rupture and coalescence. By theoretical analysis, it is shown that M-CICSAM overcomes three inherent drawbacks of the original CICSAM, concerning the basic differencing schemes, the switching strategy between different high-resolution schemes, and the far-upwind reconstruction technique on unstructured meshes. The numerical results clearly show that, compared with the existing VOF schemes of the same type, M-CICSAM produces more accurate predictions on arbitrary meshes, especially at high Courant numbers.