A Low-Dissipation Optimization-Based Gradient Reconstruction (OGRE) Scheme for Finite Volume Simulations

Finite volume methods employing second-order gradient reconstruction schemes are often utilized to computationally solve the governing equations of transport. These reconstruction schemes, while not as dissipative as first-order schemes, frequently produce either dispersive or oscillatory solutions, especially in regions of discontinuities, and/or unsatisfactory levels of dissipation in smooth regions of the variable field. A novel gradient reconstruction scheme is presented in this work which shows significant improvement over traditional second-order schemes. This Optimization-based Gradient REconstruction (OGRE) scheme works to minimize an objective function based on the mismatch between local reconstructions at midpoints between cell stencil neighbors, i.e. the degree to which the projected values of a dependent variable and its gradients in a given cell differ from each of these values in neighbor cells. An adjustable weighting parameter is included in the definition of the objective function that allows the scheme to be tuned towards greater accuracy or greater stability. This scheme is implemented using the User Defined Function capability available in the commercially available CFD solver, Ansys FLUENT. Various test cases are presented that demonstrate the ability of the new method to calculate superior predictions of both a scalar transported variable and its gradients. These cases include calculation of a discontinuous variable field, several sinusoidal variable fields and a non-uniform velocity field. Results for each case are determined on both structured and unstructured meshes, and the scheme is compared with existing standard first- and second-order upwind discretization methods.