Proposition of modified convection boundedness criterion and its evaluation for the development of bounded schemes

摘要

Numerical inaccuracies imbibe into numerical solutions depending upon the order of interpolation schemes. These schemes can be either lower-order, higher-order (HO), or very-higher-order (VHO) based on its functional relationship with control volume grid points. The conventional convection boundedness criterion (CBC) binds these interpolation schemes within the monotonic range of normalized variable diagram (NVD). Regions outside this range are approximated with the first-order upwind scheme, which reduces the numerical accuracy of interpolation schemes. These shortcomings are prevailed over by different boundedness criteria namely, extended CBC (ECBC), refinement of CBC (RCBC) and BAIR conditions. These criteria differ according to the conditions/regions enclosed by monotonic and non-monotonic ranges of NVD. Majority of these conditions extend into the non-monotonic ranges of NVD, as these bound the regions between upwind and second-order central-difference schemes. Moreover, none of the previous works detailed the importance of monotonic/non-monotonic ranges of NVD. Thus, to evaluate its importance, this work proposes a new boundedness criterion which is called as modified convection boundedness criterion (MCBC). The article further presents a detailed and in-depth comparative study of different boundedness criteria for pure advection and incompressible fluid flow tests. Results of pure advection tests suggest that, both monotonic and non-monotonic ranges of NVD play an important role in accurate advection of profiles. However, results for flow in a lid-driven square cavity suggest that, either of these boundedness criteria deliver a comparable numerical solutions. And results for flow in a backward-facing step suggest that, only CBC bounded HO schemes deliver stable numerical solutions, when compared with other criteria. In broad - spectrum, this study reveals that the numerical stability of bounded interpolation schemes rely heavily on non-monotonic ranges of NVD.