A generalized formulation for gradient reconstruction schemes in unstructured finite volume method

We present a generic framework for gradient reconstruction schemes on unstructured meshes using the notion of a dyadic sum-vector product. The proposed formulation reconstructs centroidal gradients of a scalar from its directional derivatives along specific directions in a suitably defined neighbourhood. We show that existing gradient reconstruction schemes can be encompassed within this framework by a suitable choice of the geometric vectors that define the dyadic sum tensor. The proposed framework also allows us to re-interpret some hybrid gradient schemes which are possibly not derivable through standard approaches. We also illustrate how this framework can help devise flexible gradient schemes that can enhance robustness of existing consistent gradient reconstruction approaches. Numerical results using a flexible modified Green-Gauss gradient reconstruction scheme (referred to as MGG(alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document})) on unstructured meshes and for Euler simulations are presented and a generalized approach towards flexible gradient approaches is also discussed. This study shows that a simpler tensor identity can help to unify apparently distinct gradient computation methods in a single framework and therefore must be construed as an effort towards a generalization of gradient schemes that can allow for the development and comparison of novel approaches for unstructured finite volume computations.